Wave description | PhysicsRank

A pulse travels down a rope while each small piece of rope mainly moves across it. The travelling pattern and the moving material are different things. Keeping track of both is the first step toward understanding wave equations.

We will connect those viewpoints before combining waves. That makes propagation direction, phase differences and power relations consequences of a physical model rather than a collection of signs to memorize.

What you will learn to do

Extract wavelength, frequency and direction from a travelling-wave equation.
Distinguish wave speed from the velocity and acceleration of a medium element.
Predict how string tension and linear density affect propagation.
Combine coherent waves with unequal amplitudes and nonzero phase difference.
Include source phase as well as path difference.
Compute travelling-wave power and distinguish coherent from incoherent intensity addition.

Useful before this chapterOscillations: restoring forces, phase and energy →Rates and accumulation: calculus for physics →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Travelling waves, superposition and energy
Learning goals:
- Extract wavelength, frequency and direction from a travelling-wave equation.
- Distinguish wave speed from the velocity and acceleration of a medium element.
- Predict how string tension and linear density affect propagation.
- Combine coherent waves with unequal amplitudes and nonzero phase difference.
- Include source phase as well as path difference.
- Compute travelling-wave power and distinguish coherent from incoherent intensity addition.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

Does a mark painted on a rope have to travel along the rope with a transverse pulse?

Compare your reasoning

No. The disturbance travels along the rope while the marked material moves mainly transversely in the ideal model. Wave speed and particle velocity describe different motions.

Revisit the explanation →

01 / Travelling waves, superposition and energy

A travelling pattern is different from a travelling piece of matter

Lift and lower one end of a stretched string. A disturbance can travel along the string while each small piece mainly moves up and down near its own equilibrium position. The wave speed describes the progress of the pattern; the particle velocity describes the motion of a particular piece of the medium. These velocities generally have different magnitudes and directions.

In a transverse wave, particle displacement is perpendicular to propagation. In a longitudinal wave, it is parallel. Sound in a fluid is longitudinal: local compression and expansion travel through the fluid while particles oscillate about their positions in the small-amplitude model.

A pulse is a localized disturbance; a periodic wave repeats. A sinusoidal wave is a particularly useful periodic pattern because every fixed position undergoes harmonic motion. General waveforms can be more complicated, and a medium may disperse different frequency components at different speeds.

A mechanical wave requires restoring interactions and inertia in a medium. Its propagation carries energy without requiring the medium as a whole to travel along with the pattern. Real fluids may also have bulk flow or nonlinear drift, but those are separate effects from the elementary linear-wave description.

STOP & THINK

A small mark on a string moves vertically while a pulse travels horizontally. Which statement is correct?

The mark travels horizontally at the pulse speed
Particle motion and wave propagation are different motions
The pulse cannot carry energy

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Travelling waves, superposition and energy
Focus: A travelling pattern is different from a travelling piece of matter
Learning goals:
- Extract wavelength, frequency and direction from a travelling-wave equation.
- Distinguish wave speed from the velocity and acceleration of a medium element.
- Predict how string tension and linear density affect propagation.
- Combine coherent waves with unequal amplitudes and nonzero phase difference.
- Include source phase as well as path difference.
- Compute travelling-wave power and distinguish coherent from incoherent intensity addition.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Travelling waves, superposition and energy

Follow a fixed feature to determine the propagation direction

Suppose the shape at time zero is y = f(x). The function y(x,t) = f(x − ct) shifts the same shape toward positive x at speed c. To follow a particular feature, keep its argument x − ct constant; then x increases as ct. Replacing the minus sign by a plus sign gives motion toward negative x.

This constant-phase reasoning works for a sinusoid: y = A sin(kx − ωt + φ) travels toward positive x when k and ω are positive. Its wave speed is c = ω/k. The symbol y denotes displacement of the medium, not the position of a crest along the propagation axis.

The wave number k = 2π/λ measures spatial phase change per metre. Angular frequency ω = 2πf measures temporal phase change per second. Wavelength λ and period T are different kinds of repetition: one is measured across space at an instant, the other across time at a fixed position.

All arguments of trigonometric functions must be dimensionless. If x is in metres and t in seconds, k has units rad/m and ω has units rad/s. A formula with centimetres requires consistent coefficients or a unit conversion before calculation.

y(x,t)=A\sin(kx-\omega t+\phi),\quad k=\frac{2\pi}{\lambda},\quad \omega=2\pi f,\quad c=\frac\omega k=f\lambda

These relations describe a sinusoidal travelling wave. In a dispersive medium, the phase speed c can depend on frequency.

WORKED EXAMPLE

Read a wave equation without mixing its parameters

A string displacement is y = 0.0030 sin(5πx − 40πt), with x and y in metres and t in seconds. Find amplitude, wavelength, frequency and propagation speed.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Travelling waves, superposition and energy
Focus: Follow a fixed feature to determine the propagation direction
Learning goals:
- Extract wavelength, frequency and direction from a travelling-wave equation.
- Distinguish wave speed from the velocity and acceleration of a medium element.
- Predict how string tension and linear density affect propagation.
- Combine coherent waves with unequal amplitudes and nonzero phase difference.
- Include source phase as well as path difference.
- Compute travelling-wave power and distinguish coherent from incoherent intensity addition.
Equations in this section (LaTeX):
y(x,t)=A\sin(kx-\omega t+\phi),\quad k=\frac{2\pi}{\lambda},\quad \omega=2\pi f,\quad c=\frac\omega k=f\lambda
Meaning: These relations describe a sinusoidal travelling wave. In a dispersive medium, the phase speed c can depend on frequency.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

03 / Travelling waves, superposition and energy

Differentiate at a fixed position to find the medium’s motion

To find the transverse velocity of a string element, hold its equilibrium coordinate x fixed and differentiate displacement with respect to time. For the positive-travelling sinusoid, v_y = −Aω cos(kx − ωt + φ). Differentiating again gives a_y = −ω²y.

The maximum particle speed is Aω, not ω/k. A small-amplitude wave can travel rapidly while its particles move relatively slowly. The particle speed is zero at a crest or trough of a sinusoidal snapshot, even though the crest itself continues propagating.

For any shape y = f(x − ct), the temporal derivative equals −c times the spatial slope. At a point with positive slope in a right-travelling wave, the particle moves downward. Visualizing the shape shifting right over a fixed marked position explains the sign.

The relation between derivative and slope assumes a waveform translating without changing shape. If several waves overlap, or if dispersion changes the shape, apply the relation to each appropriate component or use the full time derivative instead.

v_y=\frac{\partial y}{\partial t}=-c\frac{\partial y}{\partial x},\qquad a_y=-\omega^2y

The slope relation is for a right-travelling shape f(x − ct). For a left-travelling shape the sign reverses.

WORKED EXAMPLE

Use a snapshot to predict a marked point’s motion

A right-travelling pulse moves at 6.0 m/s. At a marked string position, the instantaneous slope ∂y/∂x is −0.020. Find the transverse velocity at that point.

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STOP & THINK

A sinusoidal wave has amplitude 0.004 m and angular frequency 50 rad/s. What is the maximum transverse particle speed?

Answer in m/s.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Travelling waves, superposition and energy
Focus: Differentiate at a fixed position to find the medium’s motion
Learning goals:
- Extract wavelength, frequency and direction from a travelling-wave equation.
- Distinguish wave speed from the velocity and acceleration of a medium element.
- Predict how string tension and linear density affect propagation.
- Combine coherent waves with unequal amplitudes and nonzero phase difference.
- Include source phase as well as path difference.
- Compute travelling-wave power and distinguish coherent from incoherent intensity addition.
Equations in this section (LaTeX):
v_y=\frac{\partial y}{\partial t}=-c\frac{\partial y}{\partial x},\qquad a_y=-\omega^2y
Meaning: The slope relation is for a right-travelling shape f(x − ct). For a left-travelling shape the sign reverses.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

04 / Travelling waves, superposition and energy

The medium sets the wave speed in the ideal string model

A curved string segment has a net transverse force because the tensions at its ends point in slightly different directions. For small slopes, constant tension and uniform linear mass density, Newton’s law leads to the wave equation with c² = F_T/μ. Here μ is mass per unit length, not total mass.

Greater tension gives a stronger restoring response to the same curvature, increasing speed. Greater mass per length increases inertia, reducing speed. The result assumes small transverse displacements, a flexible string and negligible changes of tension during the motion.

The driving source sets the frequency of a steady wave. When that wave enters a different section of string, the boundary oscillates at the same driving frequency, while the new medium speed changes the wavelength. Speed, frequency and wavelength cannot all be chosen independently.

At an interface, reflected and transmitted waves may both occur. Frequency continuity does not mean equal amplitudes or equal power. Those require force and displacement boundary conditions, not just c = fλ.

c=\sqrt{\frac{F_T}{\mu}},\qquad [\mu]=\mathrm{kg/m}

The units under the square root are (kg·m/s²)/(kg/m) = m²/s².

WORKED EXAMPLE

Predict what changes across a string junction

Two ideal string sections have the same tension. The second has four times the linear density of the first. A steady wave of frequency 25 Hz travels in the first with speed 10 m/s. Find the transmitted frequency, speed and wavelength in the second.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Travelling waves, superposition and energy
Focus: The medium sets the wave speed in the ideal string model
Learning goals:
- Extract wavelength, frequency and direction from a travelling-wave equation.
- Distinguish wave speed from the velocity and acceleration of a medium element.
- Predict how string tension and linear density affect propagation.
- Combine coherent waves with unequal amplitudes and nonzero phase difference.
- Include source phase as well as path difference.
- Compute travelling-wave power and distinguish coherent from incoherent intensity addition.
Equations in this section (LaTeX):
c=\sqrt{\frac{F_T}{\mu}},\qquad [\mu]=\mathrm{kg/m}
Meaning: The units under the square root are (kg·m/s²)/(kg/m) = m²/s².

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

05 / Travelling waves, superposition and energy

A longitudinal wave compresses neighbouring fluid elements

For a transverse wave on a string, the displacement is across the propagation direction. In a longitudinal fluid wave, particle displacement is along it. Neither statement means each particle travels with the wave crest: the material oscillates locally while the disturbance moves through it.

Picture two nearby marked fluid elements originally separated by dx. Their displacements are ξ(x) and ξ(x + dx). Their separation changes by approximately (∂ξ/∂x)dx. A negative displacement gradient squeezes the interval; a positive gradient stretches it. This is why the local pressure change depends on the spatial gradient, not on the displacement itself.

The bulk modulus B relates a small fractional compression to a pressure increase: δp = −B ∂ξ/∂x in this one-dimensional model. For ξ = A cos(kx − ωt), the pressure variation is BkA sin(kx − ωt). Displacement and pressure are a quarter cycle apart; maximum displacement is not maximum compression.

Combining that restoring response with Newton’s law for a small fluid slice gives c² = B/ρ. Use the modulus appropriate to the acoustic process; ordinary small sound disturbances in gases are approximately adiabatic, giving B = γp and c = √(γp/ρ). This collective wave speed differs from both particle oscillation speed and rms thermal molecular speed.

\delta p=-B\frac{\partial\xi}{\partial x},\qquad c=\sqrt{\frac B\rho}

ξ is longitudinal displacement in metres, B is the appropriate bulk modulus in Pa, ρ is equilibrium density in kg/m³ and c is propagation speed in m/s. The gradient is dimensionless.

WORKED EXAMPLE

Read pressure from a spatial displacement gradient

At an instant, fluid displacement decreases by 0.20 mm across an original interval of 0.40 m. The gradient is approximately uniform there and B = 3.0×10⁶ Pa. Find the pressure perturbation.

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WORKED EXAMPLE

Use stiffness and inertia to predict a wave speed

A fluid has the appropriate acoustic bulk modulus 2.0×10⁹ Pa and density 1250 kg/m³. Find its small-amplitude sound speed.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

STOP & THINK

At a spatial maximum of sinusoidal longitudinal displacement, what is the pressure perturbation in the linear model?

Zero
Maximum positive
Maximum negative

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Travelling waves, superposition and energy
Focus: A longitudinal wave compresses neighbouring fluid elements
Learning goals:
- Extract wavelength, frequency and direction from a travelling-wave equation.
- Distinguish wave speed from the velocity and acceleration of a medium element.
- Predict how string tension and linear density affect propagation.
- Combine coherent waves with unequal amplitudes and nonzero phase difference.
- Include source phase as well as path difference.
- Compute travelling-wave power and distinguish coherent from incoherent intensity addition.
Equations in this section (LaTeX):
\delta p=-B\frac{\partial\xi}{\partial x},\qquad c=\sqrt{\frac B\rho}
Meaning: ξ is longitudinal displacement in metres, B is the appropriate bulk modulus in Pa, ρ is equilibrium density in kg/m³ and c is propagation speed in m/s. The gradient is dimensionless.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

06 / Travelling waves, superposition and energy

Add displacements before reasoning about the combined wave

In a linear medium, the net displacement at a given place and time is the algebraic sum of the displacements that each wave would produce there. Two pulses can overlap and then continue travelling; the temporary combined shape does not mean the pulses have collided like rigid objects.

Signs matter. An upward displacement and an equal downward displacement cancel at that instant and location. The medium can still have velocity or stored strain elsewhere. Zero displacement alone is not proof of zero wave energy.

For sinusoidal waves of the same frequency and direction, the relative phase remains fixed in a uniform nondispersive medium. Their resultant is another sinusoid, with amplitude determined by the two amplitudes and phase difference. If frequencies differ, the phase difference changes with time and the resultant amplitude can vary.

A path difference Δr contributes a phase difference 2πΔr/λ. Include any initial source phase difference or reflection phase change as well. A path difference by itself does not establish constructive interference unless the source phases are specified.

R^2=A_1^2+A_2^2+2A_1A_2\cos\delta

This phasor result applies to two same-frequency oscillations at the observation point, with a stable relative phase δ.

WORKED EXAMPLE

Combine unequal waves at a nontrivial phase difference

Two coherent waves at an observation point have displacement amplitudes 3.0 mm and 4.0 mm and phase difference π/2. Find the resultant amplitude.

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STOP & THINK

Two coherent waves of amplitudes 3 mm and 5 mm have opposite phase at a point. Can their resultant amplitude be zero?

Yes, opposite phase always cancels completely
No, the remaining amplitude is 2 mm
No, the remaining amplitude is 8 mm

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Travelling waves, superposition and energy
Focus: Add displacements before reasoning about the combined wave
Learning goals:
- Extract wavelength, frequency and direction from a travelling-wave equation.
- Distinguish wave speed from the velocity and acceleration of a medium element.
- Predict how string tension and linear density affect propagation.
- Combine coherent waves with unequal amplitudes and nonzero phase difference.
- Include source phase as well as path difference.
- Compute travelling-wave power and distinguish coherent from incoherent intensity addition.
Equations in this section (LaTeX):
R^2=A_1^2+A_2^2+2A_1A_2\cos\delta
Meaning: This phasor result applies to two same-frequency oscillations at the observation point, with a stable relative phase δ.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

07 / Travelling waves, superposition and energy

Use the full phase difference to predict interference

For two equal-frequency sources, write the phase reaching the observation point from each source and subtract. This approach keeps source phase and propagation phase in one equation and prevents memorized path rules from being used outside their conditions.

If the sources are in phase and the propagation paths share the same wavelength, path differences of integer wavelengths give constructive interference, while half-odd-integer wavelengths give destructive interference for equal amplitudes. If the sources begin in opposite phase, those roles reverse.

In a real measurement, stable interference requires sufficient coherence over the observation time. If the relative phase wanders rapidly, the average cross term may vanish. Then intensities add even though instantaneous displacements still obey superposition.

WORKED EXAMPLE

Account for source phase as well as travel distance

Two coherent sources emit equal-amplitude waves with the same wavelength. They start in opposite phase. At a point, their path lengths differ by λ/2. Predict the interference.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Travelling waves, superposition and energy
Focus: Use the full phase difference to predict interference
Learning goals:
- Extract wavelength, frequency and direction from a travelling-wave equation.
- Distinguish wave speed from the velocity and acceleration of a medium element.
- Predict how string tension and linear density affect propagation.
- Combine coherent waves with unequal amplitudes and nonzero phase difference.
- Include source phase as well as path difference.
- Compute travelling-wave power and distinguish coherent from incoherent intensity addition.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

08 / Travelling waves, superposition and energy

Energy transport depends on amplitude and frequency as well as the medium

A travelling wave on a string carries kinetic energy of moving elements and elastic energy associated with their deformation. For a small-amplitude sinusoidal wave, the average kinetic and elastic contributions are equal. Their sum gives a mean energy per length of ½μω²A².

Multiplying mean energy per length by propagation speed gives mean transmitted power: P̄ = ½μω²A²c. This is a travelling-wave result under the same ideal string assumptions. It is not a formula for assigning a one-way power to a standing wave without considering its oppositely travelling components.

At fixed medium and frequency, doubling amplitude multiplies average power by four. At fixed amplitude and nondispersive speed, doubling frequency also multiplies it by four. These comparisons hold only when the stated quantities really are fixed; changing tension can change the medium speed at the same time.

Intensity is power per area normal to propagation. It is useful for waves spreading through space, while power is usually the convenient quantity for a wave carried along one string. For an isotropic point source with no absorption or reflection, its fixed power spreads over spheres of area 4πr², giving I = P/(4πr²).

\overline P=\frac12\mu\omega^2A^2c,\qquad I=\frac{P}{\mathcal A}

The first expression is for a sinusoidal travelling wave on an ideal string. The area in the intensity definition is perpendicular to the energy-flow direction.

WORKED EXAMPLE

Calculate power and check the units

A sinusoidal travelling wave has μ = 0.020 kg/m, c = 10 m/s, angular frequency 100 rad/s and amplitude 2.0 mm. Find its mean power.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Travelling waves, superposition and energy
Focus: Energy transport depends on amplitude and frequency as well as the medium
Learning goals:
- Extract wavelength, frequency and direction from a travelling-wave equation.
- Distinguish wave speed from the velocity and acceleration of a medium element.
- Predict how string tension and linear density affect propagation.
- Combine coherent waves with unequal amplitudes and nonzero phase difference.
- Include source phase as well as path difference.
- Compute travelling-wave power and distinguish coherent from incoherent intensity addition.
Equations in this section (LaTeX):
\overline P=\frac12\mu\omega^2A^2c,\qquad I=\frac{P}{\mathcal A}
Meaning: The first expression is for a sinusoidal travelling wave on an ideal string. The area in the intensity definition is perpendicular to the energy-flow direction.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

09 / Travelling waves, superposition and energy

Interference redistributes energy; it does not repeal conservation

For coherent waves reaching a point in the same kind of medium, with compatible polarization and propagation geometry, intensity is proportional to squared resultant amplitude. The interference term can therefore make the local intensity greater or smaller than the sum of the separate intensities.

For equal individual intensities I₀, the in-phase local result is 4I₀ and the opposite-phase result can be zero. This does not mean two independent powers have globally become four or vanished. Interference changes the spatial distribution and, depending on the source arrangement, can change the work required to drive the sources.

For independent incoherent sources averaged over many relative phases, the average cross term vanishes and intensities add. Do not use amplitude addition for incoherent average intensities, or add coherent intensities before accounting for phase.

A reliable workflow is to identify the observable first. For displacement, add signed displacements. For a same-frequency resultant, combine amplitudes with phase. For power or intensity, square the appropriate amplitude or use the energy-flow law after the interference calculation.

STOP & THINK

Two equal coherent waves each produce intensity I₀ at a point when acting alone. Their received phase difference is π/2. What is the resultant intensity divided by I₀?

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Travelling waves, superposition and energy
Focus: Interference redistributes energy; it does not repeal conservation
Learning goals:
- Extract wavelength, frequency and direction from a travelling-wave equation.
- Distinguish wave speed from the velocity and acceleration of a medium element.
- Predict how string tension and linear density affect propagation.
- Combine coherent waves with unequal amplitudes and nonzero phase difference.
- Include source phase as well as path difference.
- Compute travelling-wave power and distinguish coherent from incoherent intensity addition.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

A fixed phase tracks the wave pattern; a time derivative at fixed position tracks the medium.
For f(x − ct), particle velocity equals minus c times the local slope.
An ideal string has c = √(F_T/μ), while a steady source sets frequency.
Superposition adds displacements, including signs and phase.
Complete destructive interference needs equal received amplitudes as well as opposite phase.
Travelling-wave power scales as A²ω² in a fixed ideal string.
Coherent interference changes local intensity through a phase-dependent cross term.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

A sinusoidal wave has k = 8 rad/m and ω = 120 rad/s. What is its phase speed?

Answer in m/s.

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SELF CHECK 2

At fixed string properties and frequency, amplitude becomes 1.5 times as large. What is the new-to-old mean-power ratio?

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SELF CHECK 3

For positive k and ω, which wave travels toward negative x?

A sin(kx − ωt)
A sin(kx + ωt)
Both travel toward positive x

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SELF CHECK 4

A sinusoidal travelling wave has a crest at a marked string position at an instant. What is that element’s transverse velocity at that instant?

Zero
Equal to the wave speed
Maximum in magnitude

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 5

For independent incoherent sources averaged over rapidly varying relative phase, which quantities add?

Their signed amplitudes as fixed phasors
Their average intensities
Their frequencies

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APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

Read a wave from its equationIntermediate · Free sample · Hints and guided reasoning A pressure disturbance has a bulk elastic responseIntermediate · Hints and guided reasoning Compression is a spatial change of displacementAdvanced · Hints and guided reasoning Combine amplitudes with phaseIntermediate · Hints and guided reasoning Trade amplitude against frequencyIntermediate · Hints and guided reasoning A string point moves differently from the waveIntermediate · Hints and guided reasoning

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

How can you determine the direction of a wave written as f(x − vt) without memorizing a sign slogan?

Compare your reasoning

Track a fixed feature by holding x − vt constant. As time increases, its position x increases when v is positive. The pattern therefore travels in the positive x direction.

Revisit the explanation →

A few days later: change the situation

A longitudinal displacement is large but locally the same for neighbouring particles. Must the pressure perturbation be large?

Compare your reasoning

No. Compression depends on the spatial displacement gradient. Neighbours moving together can translate without changing their separation. In the linear model δp = −B ∂ξ/∂x.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

Back to chapter contents ↑