Wave Motion

A progressive (travelling) wave is a disturbance that moves continuously forward through a medium in a definite direction with constant velocity, transferring energy from particle to particle without any net transport of matter.

Derivation: let the particle at the origin (\(x=0\)) execute SHM, \(y=A\sin\omega t\). For a wave travelling in the +x direction with velocity \(v\), the disturbance takes a time \(t’=\dfrac xv\) to reach a point at distance \(x\). So the displacement at \(x\) at time \(t\) equals that of the origin at the earlier time \(t-t’\):

Using \(\omega=\dfrac{2\pi}{T}\) and \(v=\dfrac\lambda T\), we get \(\dfrac\omega v=\dfrac{2\pi}{\lambda}=k\), the angular wave number:

Different forms:

For a wave travelling in the \(-x\) direction, replace \(-kx\) by \(+kx\).

Sound waves travelling through air, and light (electromagnetic) waves travelling from the sun to the earth.

The phase \(kx-\omega t\) stays constant when \(x\) increases with \(t\), so this is also a wave moving in the +x direction. It differs from \(A\sin(\omega t-kx)\) only by a sign, that is, a phase difference of \(\pi\).

\(y(x,t)\): the displacement of the particle at position \(x\) and time \(t\) from its equilibrium position.

\(A\) (amplitude): the maximum displacement of the particles from equilibrium.

\(k\) (angular wave number): \(k=\dfrac{2\pi}{\lambda}\), the number of radians of phase per unit distance.

\(\omega\) (angular frequency): \(\omega=2\pi f\), the rate of oscillation in radians per unit time.

\(x\) and \(t\): the position along the direction of propagation and the time elapsed.

Comparing with \(y=A\sin(kx-\omega t)\): \(A=0.02\), \(k=5\), \(\omega=100\).

(i) Amplitude: \(A=0.02\ \mathrm{m}\)

(ii) Wavelength:

(iii) Frequency:

(iv) Speed:

The two waves must have the same amplitude \(A\), the same frequency and wavelength, and must travel in exactly opposite directions along the same line in a bounded medium.

A stationary (standing) wave is the interference pattern formed by the superposition of two identical progressive waves travelling in opposite directions. The waveform does not move forward; it has fixed points of zero displacement (nodes) and of maximum displacement (antinodes), and the energy stays confined in the medium.

Examples: transverse standing waves on a plucked guitar string, and longitudinal standing waves in the air column of a flute or organ pipe.

Here \(k=4\ \mathrm{rad\,m^{-1}}\) and \(\omega=200\ \mathrm{rad\,s^{-1}}\).

(i) Equation of the stationary wave: by superposition and the identity \(\sin(a-b)+\sin(a+b)=2\sin a\cos b\):

(ii) Distance between consecutive nodes:

When two or more waves travel through the same medium at the same time, the resultant displacement of any particle at any instant is the sum of the displacements that each wave would produce independently:

When a forward wave and a backward (reflected) wave cross, their displacements add. At some points a crest of one wave always meets a trough of the other, giving permanent destructive interference and zero displacement: these are the nodes. Midway between the nodes, crest meets crest and trough meets trough, giving permanent constructive interference and maximum displacement: these are the antinodes.

Let the waves be \(y_1=A\sin(\omega t-kx)\) (towards +x) and \(y_2=A\sin(\omega t+kx)\) (towards \(-x\)). Then \(y=y_1+y_2\). Using \(\sin C+\sin D=2\sin\dfrac{C+D}{2}\cos\dfrac{C-D}{2}\):

Since \(\cos(-kx)=\cos kx\):

The factor \(2A\cos kx\) is the position-dependent amplitude. (With another choice of initial phase the same wave can be written \(2A\sin kx\cos\omega t\).)

A string fixed at both ends with 3 antinodes has 3 loops, each equal to \(\dfrac\lambda2\):

(i) Wavelength:

(ii) Wave speed:

Plucking a guitar string sends transverse progressive waves toward both ends, which are clamped at the bridge and the nut. At these fixed ends the waves reflect. The incident wave and the reflected wave have the same frequency and amplitude and travel in opposite directions, so they superpose. The fixed ends are forced to be nodes, so only certain resonant frequencies survive and form standing waves.

In \(y=[2A\sin kx]\cos\omega t\), the bracket \(2A\sin kx\) is the amplitude of the particle at position \(x\).

Nodes: where \(\sin kx=0\) the amplitude is always zero. These points never move and are the places of maximum strain (or pressure change).

Antinodes: where \(\sin kx=\pm1\) the amplitude is the maximum, \(2A\). These points oscillate most vigorously and have minimum strain.

A stationary wave is the sum of two waves carrying equal energy in opposite directions. The forward transport of energy by one wave is cancelled by the backward transport by the other, so the net flow of energy is zero. With no net transfer of energy, the pattern stays in place and the particles simply oscillate where they are.

In the second harmonic the string has 2 loops: \(L=2\times\dfrac\lambda2=\lambda\).

(ii) Wavelength: \(\lambda=L=1.5\ \mathrm{m}\)

(i) Speed:

Nodes: the amplitude \(|2A\cos kx|\) is zero when \(\cos kx=0\), that is \(kx=(2n+1)\dfrac\pi2\), \(n=0,1,2,\dots\) With \(k=\dfrac{2\pi}{\lambda}\):

Antinodes: the amplitude is maximum (\(2A\)) when \(\cos kx=\pm1\), that is \(kx=n\pi\):

Distance between successive nodes: the nodes with \(n=0\) and \(n=1\) are at \(\dfrac\lambda4\) and \(\dfrac{3\lambda}{4}\):

Stationary waves need two waves of the same frequency, wavelength and amplitude travelling in opposite directions in a bounded medium.

Stretched string: a plucked string sends transverse waves to its fixed ends, where displacement must be zero (a node). The reflected wave superposes with the incoming wave and a standing wave forms when the string holds a whole number of half wavelengths, \(L=n\dfrac\lambda2\).

Air column (pipe closed at one end): a longitudinal sound wave travels down the tube and reflects. The closed rigid end is a displacement node, because the air cannot move past the barrier, and the open end is an antinode. The incident and reflected sound waves superpose to form a standing longitudinal wave.

(i) Wavelength:

(ii) Equation of the wave: use \(y=A\sin(\omega t-kx)\) with