Superposition and Reflection

Why this chapter

Why this chapter exists

A single wave on an endless string is a clean idea, but no real wave is alone and no real string is endless. Waves meet other waves, and they reach the ends of what carries them. This chapter answers what happens then. Overlapping waves simply add, and then go on as if nothing had happened. At an end, the string creates a new wave to keep the end's rule, and that is what we call a reflection. These two ideas together build the standing waves of the next chapter, and the sound of every string instrument.

You will learn

By the end of this chapter you will

State the principle of superposition, say when it holds, and add displacements with their signs, from a graph or a table.
Explain why overlapping waves pass through each other unchanged.
Define interference and coherent sources, and relate the phase difference to the path difference.
State the conditions for constructive and destructive interference, and find the resultant of two equal waves, \(2A\cos(\Delta\varphi/2)\), with its intensity.
Explain why a reflection is a new wave, and prove that it obeys the wave equation.
Derive the inverted reflection at a fixed end and the upright reflection at a free end, and explain the double amplitude at a free end.
Predict the reflected and transmitted pulses at a junction of two strings, and find their amplitude ratios and power shares.

Before you begin

From the chapter Introduction to Waves, you should know the travelling wave \(y = f(x - vt)\), the harmonic wave \(y = A\sin(kx - \omega t)\), the wave equation, and the relation \(v_{\text{particle}} = -v \times \text{slope}\). From the chapter Propagating Waves on a String, you should know \(v = \sqrt{T/\mu}\) and the power of a harmonic wave, \(\tfrac12\mu v\omega^2A^2\). You should be able to use the chain rule and the identity for \(\sin C + \sin D\).

Already know this?

Lessons

The Principle of Superposition

When waves overlap in a linear medium, the displacement of every particle is the sum of the displacements each wave would give alone. Two pulses on a string, a table of numbers and two laser beams show the rule, and the waves leave unchanged.

Superposition in a linear mediumSuperposition read from a tableWaves pass through unchanged

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Interference of Waves on a String

Combining waves by superposition is interference. Coherent sources keep a constant phase difference, and that phase difference, often set by a path difference, decides whether the waves reinforce or cancel. Two harmonic waves travelling the same way give an amplitude \(2A\cos(\Delta\phi/2)\).

Interference and coherenceConstructive and destructive interferenceTwo harmonic waves with a phase difference

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Reflection Makes a New Wave

A reflection is not a bounce. The state of a string is its displacement and its slope, and a boundary imposes a rule on one of them. The disturbance the boundary creates to keep its rule must itself obey the wave equation, so it is a wave, the reflected wave.

Reflection is not a bounceDisplacement slope and the boundary rulesThe new disturbance is a wave

14 min0%

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Fixed and Free Ends

At a fixed end the reflected wave must cancel the incident wave at the end, so it returns inverted, with a phase change of \(\pi\). At a free end it must cancel the slope, so it returns upright, and the end swings to twice the amplitude.

Reflection at a fixed endReflection at a free endDouble amplitude at a free end

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Transmission at a Junction

Where two strings of different mass density meet, part of a wave is reflected and part is transmitted, at the same frequency. Light to heavy acts like a rigid end, heavy to light like a free end. Continuity at the junction gives the amplitude ratios, and the two powers add to the incident power.

Frequency is unchanged at a junctionLight to heavy and heavy to lightAmplitude ratios at a junction

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

Glossary

The key words of this chapter, with the meaning this chapter gives them.

amplitude reflection ratio: The ratio \(r = A_r/A_i = (v_2 - v_1)/(v_1 + v_2)\) of the reflected to the incident amplitude at a junction. A negative value means an inverted reflection.
amplitude transmission ratio: The ratio \(t = A_t/A_i = 2v_2/(v_1 + v_2)\) of the transmitted to the incident amplitude at a junction. It is always positive.
boundary: A special point of a medium, such as the end of a string, that imposes an outside rule on the motion there.
boundary condition: The rule a boundary imposes at all times: \(y = 0\) at a rigid boundary, \(\partial y/\partial x = 0\) at a free boundary.
coherent sources: Sources that emit waves of the same frequency with a phase difference that stays constant in time. Only they give a stable interference pattern.
constructive interference: Interference in which the waves arrive in phase, with a phase difference of \(0, 2\pi, 4\pi, \dots\), so that their amplitudes add.
destructive interference: Interference in which the waves arrive out of phase, with a phase difference of \(\pi, 3\pi, 5\pi, \dots\), so that their amplitudes subtract, down to complete cancellation.
free boundary: A boundary that holds the slope of the string at zero, such as a massless ring sliding on a smooth rod. It is also called a free end.
hard reflection: A reflection with a phase change of \(\pi\), so that the pulse returns inverted. It happens at a fixed end, or from a lighter string to a heavier one.
intensity: The power carried by a wave per unit area across its direction of travel, in W/m². It grows as the square of the amplitude.
interference: The combining of two or more waves according to the principle of superposition.
junction: The point where two strings of different linear mass density are joined. A wave reaching it is partly reflected and partly transmitted.
linear medium: A medium whose waves obey a linear wave equation, so that the sum of two waves is again a wave. A string or air behaves this way when the amplitude is too small to change the medium.
method of images: A way of finding a reflected wave. The string is imagined to continue beyond the boundary, with an image pulse placed there: inverted for a fixed end, upright for a free end. The boundary rule then holds at every instant.
path difference: The difference \(\Delta x\) between the distances two waves travel to reach a point. It gives a phase difference \(\Delta\varphi = (2\pi/\lambda)\Delta x\).
phase difference: The difference \(\Delta\varphi\) between the phases of two waves at a point, in radians or degrees. It decides the type of interference.
principle of superposition: When waves overlap, each particle is displaced by the vector sum of the displacements the waves would give alone. On a string, \(y_{net} = y_1 + y_2 + \dots\)
reflected wave: The new wave that a boundary creates so that its rule is obeyed. It obeys the wave equation and travels back into the medium.
rigid boundary: A boundary that holds the displacement at zero, such as a string tied to a wall. It is also called a fixed end.
soft reflection: A reflection with no phase change, so that the pulse returns upright. It happens at a free end, or from a heavier string to a lighter one.
transmitted wave: The part of a wave that passes through a junction into the second medium. It has the same frequency and is always upright.