Superposition, Interference and Reflection

What happens when two pulses meet?

Send an upward pulse and a downward pulse towards each other along a rope. Before meeting, each pulse has its own shape. While they overlap, there is only one rope, so a particular element has one net displacement. In the small-amplitude linear model, calculate it by adding the two displacements with their signs at that same position and instant. After overlap, the pulses emerge and continue in their original directions. A temporary complicated shape is not evidence that the pulses permanently changed each other.

If two simultaneous contributions are +3 mm and −2 mm, the net displacement is +1 mm, not 5 mm. Equal opposite pulses can momentarily give zero displacement throughout their overlap. Zero displacement at an instant does not imply zero particle velocity, and it does not establish that all mechanical energy vanished. The moving material may still carry kinetic energy. The waves can reappear after overlap because the instantaneous displacement alone is not the entire dynamical state.

For several right-moving shapes with the same speed, y = F₁(x − vt) + F₂(x − vt) + … is another translating shape. Superposition is more general: opposite propagation directions and different frequencies may also be added, but their sum need not be one rigidly translating wave. At large amplitudes the restoring force can become nonlinear, and this simple addition rule may fail.

Interference of two matching harmonic waves

Now take two continuous harmonic waves travelling in the same direction with equal amplitude a, equal k and equal ω, but with a fixed phase difference φ. Set θ = kx − ωt. Then y₁ = a sin θ and y₂ = a sin(θ + φ). Their maintained phase difference is what makes the interference stable. This comparison is different from pulses that overlap only briefly.

The frequency, wavelength and direction are unchanged because the sine still contains the same kx − ωt. The amplitude changes with the relative phase. At φ = 0, both waves raise or lower an element together and the resultant amplitude is 2a: constructive interference. More generally φ = 2mπ produces the same result. At φ = (2m + 1)π, one displacement is always opposite to the other and the resultant is zero: complete destructive interference for these equal waves.

Complete cancellation requires equal amplitudes as well as opposite phase. With unequal amplitudes, opposition leaves a residual oscillation equal to the amplitude difference. Also distinguish displacement from intensity: amplitudes add with phase; intensities are not generally added as if they were signed displacements. In the exact idealisation of two equal co-propagating waves everywhere opposite, their total displacement and velocity field can vanish. Describing two separately generated components does not create extra energy in their zero total field; energy accounting must include the sources and interference terms.

Reflection is controlled by what the boundary allows

A pulse arriving at a boundary exerts a force on it. The boundary’s response acts back on the medium and can create a reflected disturbance. At an ideal fixed end of a string, the endpoint cannot move, so the total displacement there must be zero at every time. An incident upward pulse therefore reflects as a downward pulse. At a lossless fixed boundary the reflected pulse has the same magnitude and shape with an inversion, often described as a phase reversal of π for a harmonic wave.

At an ideal free end, modelled by a string attached to a freely sliding light ring, the endpoint can move transversely. There is no net transverse force at the endpoint, so the string tangent has zero slope there in the small-slope model. An upward pulse reflects upward without inversion. At full overlap the endpoint displacement can be twice the displacement amplitude of either incident or reflected pulse. This does not mean that a free endpoint is fixed at a nonzero height: it oscillates.

Reflection equations must reverse the direction as well as satisfy the end condition

Choose the boundary at x = 0 and the string on x ≤ 0. An incident wave moving right towards the boundary is yᵢ = a cos(kx − ωt), with k,ω positive. A reflected wave must move left, so its time-dependent phase must involve kx + ωt. Choose its sign by enforcing the boundary condition. The arbitrary phase origin is chosen for clarity; it does not change pulse inversion.

As a cross-check, if instead yᵢ = a sin(kx − ωt) in this same coordinate system, the fixed-end reflected wave is +a sin(kx + ωt). At x = 0 the two terms are −a sin(ωt) and +a sin(ωt), so they cancel. The free-end reflected wave is −a sin(kx + ωt). These apparent sign changes arise from the sine phase convention, not from a change of the physical reflection rule. Use the endpoint test to prevent sign errors. Leaving kx − ωt unchanged in a supposed reflected wave fails to reverse the propagation direction.

An interface can both reflect and transmit

A perfectly fixed or perfectly free end is an ideal limiting boundary. At an interface between two real elastic media, some incident energy is generally reflected and some transmitted. For incidence at an angle, the transmitted direction may bend: refraction. The directions obey the reflection law and, in the appropriate isotropic wave model, the refraction relation sin i/sin r = v₁/v₂. Here i and r are measured from the normal, and v₁ and v₂ are the corresponding phase speeds. This geometric relation is stated for context; deriving transmission amplitudes lies outside this chapter.

For sound the phrase “phase reversal” must specify the variable. A rigid closed end is a displacement node but a pressure antinode; an ideal open end is a pressure node but a displacement antinode. A reflected pressure disturbance can therefore have the opposite sign convention to a reflected displacement disturbance at the same end. We will build this distinction into the air-column modes, rather than applying the string’s displacement rule blindly to pressure.

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