Question

A longitudinal wave has a wavelength of \(10 \mathrm{cm}\) and an amplitude of \(5.0 \mathrm{cm}\) and travels in the \(y\) -direction. The wave speed in this medium is \(80 \mathrm{cm} / \mathrm{s},\) (a) Describe the motion of a particle in the medium as the wave travels through the medium. (b) How would your answer differ if the wave were transverse instead?

Short answer

Answer: If the wave was a transverse wave, the particles in the medium would move perpendicular to the direction of the wave propagation, up and down along the x-axis, instead of parallel to the wave propagation like in a longitudinal wave.

Step-by-step solution

Part (a): Describing the motion of a particle in the medium for a longitudinal wave

In a longitudinal wave, particles move parallel to the direction of the wave's propagation. In this case, the wave is moving in the y-direction with a wavelength of 10 cm and an amplitude of 5.0 cm. The motion of the particle in the medium can be described using the equation: $$ y(t) = A \cos (kx - \omega t + \phi) $$ where \(y(t)\): Displacement of the particle \(A\): Amplitude \(k\): Wave number (equal to \(2\pi / \lambda\), \(\lambda\) being the wavelength) \(\omega\): Angular frequency (equal to \(2\pi / T\), T being the period) \(t\): Time \(x\): Position of the particle \(\phi\): Phase constant Using the given values and the speed of the wave \(v = 80 \mathrm{cm}/s\), we can find the period (\(T\)) and frequency (\(f\)) of the wave: $$ T = \frac{\lambda}{v} = \frac{10 cm}{80 cm/s} = 0.125 s $$ $$ f = \frac{1}{T} = 8 Hz $$ Now we can find the angular frequency \(\omega\): $$ \omega = 2\pi f = 16 \pi rad/s $$ And the wave number \(k\): $$ k = \frac{2\pi}{\lambda} = \frac{2\pi}{10 cm} = \frac{\pi}{5} rad/cm $$ Now we can write the motion of the particle as: $$ y(t) = 5.0 \cos \left(\frac{\pi}{5}x - 16 \pi t + \phi \right) $$ This equation describes the displacement of a particle in the medium as the longitudinal wave travels through it.

Part (b): Comparison of motion with a transverse wave

If the wave was a transverse wave instead of a longitudinal wave, the motion of particles in the medium would be perpendicular to the direction of the wave propagation (y-direction). The particles would move up and down along the x axis instead. The equation representing the motion of a particle in the medium would still be the same, but it would now represent movement in the x direction instead of in the y-direction: $$ x(t) = 5.0 \cos \left(\frac{\pi}{5}y - 16 \pi t + \phi \right) $$ In summary, the main difference in motion between longitudinal and transverse waves is the direction in which the particles in the medium move: parallel to the direction of wave propagation for longitudinal waves, and perpendicular to the direction of wave propagation for transverse waves.