Waves: common doubts

Question 1

Do the particles of the medium travel along with a wave?

Accepted Answer

No. Each particle only oscillates about its own mean position; it is the disturbance and its energy that travel. A cork on a pond bobs up and down as ripples pass but stays roughly where it is. A wave carries energy and momentum from one place to another without carrying the medium along.

Question 2

Why does sound need a medium but light does not?

Accepted Answer

Sound is a mechanical wave: it travels by particles of a medium pushing on their neighbours, so without particles there is nothing to pass the disturbance along. Light is an electromagnetic wave made of changing electric and magnetic fields, which need no material medium and can cross empty space.

Question 3

What is the difference between transverse and longitudinal waves?

Accepted Answer

In a transverse wave the particles oscillate perpendicular to the direction the wave travels, as on a plucked string. In a longitudinal wave they oscillate along the direction of travel, forming compressions and rarefactions, as in sound in air. Transverse mechanical waves need a medium that resists shearing, which is why they cannot travel through the bulk of a fluid.

Question 4

Can sound travel as a transverse wave?

Accepted Answer

In gases and liquids, no; sound there is always longitudinal, because fluids have no shear modulus to support sideways oscillations. Solids can carry both kinds of mechanical wave, transverse and longitudinal, since they resist both shearing and compression. This is why both wave types travel through solid rock.

Question 5

How do you read wavelength and frequency from a wave equation?

Accepted Answer

Compare it with y(x, t) = a sin(kx − ωt + φ). The coefficient of x is the wave number k, and λ = 2π/k, not 1/k. The coefficient of t is ω, and the frequency is ν = ω/2π. The wave speed is v = ω/k. Mixing up k with 1/λ or ω with ν is the usual mistake.

Question 6

How can you tell the direction a wave is moving from its equation?

Accepted Answer

If x and t appear with opposite signs, as in sin(kx − ωt), the wave moves in the positive x direction. If they have the same sign, as in sin(kx + ωt), it moves in the negative x direction. To see why, follow a crest: kx − ωt stays constant only if x increases as t increases.

Question 7

What is the difference between wave velocity and particle velocity?

Accepted Answer

Wave velocity, v = λν, is how fast the disturbance moves through the medium and is constant for a given medium. Particle velocity is how fast an individual particle oscillates about its mean position, ∂y/∂t, which keeps changing with time. The two are entirely different quantities and are usually very different in size.

Question 8

What decides the speed of a wave on a string?

Accepted Answer

Only the tension T and the mass per unit length µ of the string: v = √(T/µ). The speed does not depend on the frequency of the source. A higher frequency gives a shorter wavelength on the same string, so that λν stays equal to the fixed speed. Tighter or lighter strings carry waves faster.

Question 9

Why was Newton's formula for the speed of sound wrong?

Accepted Answer

Newton assumed the compressions and rarefactions of sound happen isothermally, giving v = √(P/ρ), about 280 m/s in air, well below the measured value. Laplace pointed out the changes are too fast for heat to flow, so the process is adiabatic. That brings in γ, giving v = √(γP/ρ), which matches experiment.

Question 10

Does pressure affect the speed of sound in air?

Accepted Answer

No, not at a fixed temperature. In v = √(γP/ρ), raising the pressure raises the density in the same ratio, so P/ρ stays constant. Temperature does change the speed, since warmer air at the same pressure is less dense; the speed of sound in a gas grows as the square root of absolute temperature.

Question 11

Why does sound travel faster in solids than in air?

Accepted Answer

Because wave speed depends on the ratio of an elastic modulus to density, v = √(Y/ρ) in a bar. Solids are far more resistant to compression than gases, and this large modulus outweighs their greater density. So sound travels several kilometres per second in steel but only about 340 m/s in air.

Question 12

What is the principle of superposition of waves?

Accepted Answer

When two or more waves pass through the same point, the resulting displacement is the algebraic sum of the displacements each wave would produce alone. Afterwards, each wave carries on unchanged. Superposition explains interference, standing waves and beats, which all come from adding displacements of overlapping waves.

Question 13

What happens when two waves meet in phase and out of phase?

Accepted Answer

If two identical waves meet in phase, their displacements add and the resultant amplitude is doubled, giving constructive interference. If they are exactly out of phase, by π, they cancel and the amplitude becomes zero, giving destructive interference. For any phase difference φ, the resultant amplitude is 2a cos(φ/2).

Question 14

What happens to a wave reflected from a rigid boundary?

Accepted Answer

It comes back inverted, with a phase change of π. A rigid end cannot move, so the incoming and reflected waves must add to zero displacement there, which requires the reflected pulse to be upside down. At an open or free boundary, the wave reflects without inversion, as the end is free to move.

Question 15

What is the distance between two consecutive nodes in a standing wave?

Accepted Answer

It is λ/2, half a wavelength, not a full wavelength. The distance between a node and the nearest antinode is λ/4. A standing wave does not travel: nodes stay permanently at rest and antinodes oscillate with the largest amplitude, so energy is not carried along the medium.

Question 16

What is the difference between a standing wave and a travelling wave?

Accepted Answer

A travelling wave moves through the medium, and every particle oscillates with the same amplitude, one after another. A standing wave forms when two identical waves travel in opposite directions; its pattern stays in place, with fixed nodes and antinodes, and the amplitude varies from point to point as 2a sin kx.

Question 17

Why does a closed pipe produce only odd harmonics?

Accepted Answer

A pipe closed at one end must have a node at the closed end and an antinode at the open end. The pipe's length must then be an odd number of quarter wavelengths, so the allowed frequencies are (n + ½)v/2L: v/4L, 3v/4L, 5v/4L and so on. The even multiples cannot fit these boundary conditions.

Question 18

What are the harmonics of a string fixed at both ends?

Accepted Answer

A string fixed at both ends must have a node at each end, so its length must be a whole number of half wavelengths. This gives frequencies ν = nv/2L, with n = 1, 2, 3, and so on. All harmonics are present, and the lowest, v/2L, is called the fundamental.

Question 19

What is the beat frequency?

Accepted Answer

It is the difference between the two frequencies, ν_beat = ν₁ − ν₂, not their sum. When two sounds of nearly equal frequency are heard together, their superposition makes the loudness rise and fall regularly. The number of loud swells per second equals this frequency difference.

Question 20

How are beats used to tune a musical instrument?

Accepted Answer

A musician sounds the instrument together with a reference note and listens for beats. If beats are heard, the frequencies differ; the musician adjusts the string tension until the beats slow down and finally disappear, meaning the two frequencies match. This works because beats are easy to hear even when the frequency difference is tiny.

Waves: common doubts, answered

Waves carry energy, not matter