91Ó°ÊÓ

An underground explosion sends out both transverse (S waves) and longitudinal (P waves) mechanical wave pulses (seismic waves) through the crust of the Earth. Suppose the speed of transverse waves is \(8.0 \mathrm{km} / \mathrm{s}\) and that of longitudinal waves is \(10.0 \mathrm{km} / \mathrm{s} .\) On one occasion, both waves follow the same path from a source to a detector (a seismograph); the longitudinal pulse arrives 2.0 s before the transverse pulse. What is the distance between the source and the detector?

When the string of a guitar is pressed against a fret, the shortened string vibrates at a frequency \(5.95 \%\) higher than when the previous fret is pressed. If the length of the part of the string that is free to vibrate is \(64.8 \mathrm{cm},\) how far from one end of the string are the first three frets located?

A guitar string has a fundamental frequency of \(300.0 \mathrm{Hz}\) (a) What are the next three lowest standing wave frequencies? (b) If you press a finger lightly against the string at its midpoint so that both sides of the string can still vibrate, you create a node at the midpoint. What are the lowest four standing wave frequencies now? (c) If you press hard at the same point, only one side of the string can vibrate. What are the lowest four standing wave frequencies?

(a) Write an equation for a surface seismic wave moving along the \(-x\) -axis with amplitude \(2.0 \mathrm{cm},\) period \(4.0 \mathrm{s}\) and wavelength $4.0 \mathrm{km} .\( Assume the wave is harmonic, \)x\( is measured in \)\mathrm{m},$ and \(t\) is measured in \(\mathrm{s}\). (b) What is the maximum speed of the ground as the wave moves by? (c) What is the wave speed?

The formula for the speed of transverse waves on a spring is the same as for a string. (a) A spring is stretched to a length much greater than its relaxed length. Explain why the tension in the spring is approximately proportional to the length. (b) A wave takes 4.00 s to travel from one end of such a spring to the other. Then the length is increased \(10.0 \% .\) Now how long does a wave take to travel the length of the spring? [Hint: Is the mass per unit length constant?]

Deep-water waves are dispersive (their wave speed depends on the wavelength). The restoring force is provided by gravity. Using dimensional analysis, find out how the speed of deep-water waves depends on wavelength \(\lambda\), assuming that \(\lambda\) and \(g\) are the only relevant quantities. (Mass density does not enter into the expression because the restoring force, arising from the weight of the water, is itself proportional to the mass density.)

In contrast to deep-water waves, shallow ripples on the surface of a pond are due to surface tension. The surface tension \(\gamma\) of water characterizes the restoring force; the mass density \(\rho\) of water characterizes the water's inertia. Use dimensional analysis to determine whether the surface waves are dispersive (the wave speed depends on the wavelength) or non dispersive (their wave speed is independent of wavelength). [Hint: Start by assuming that the wave speed is determined by \(\gamma, \rho,\) and the wavelength \(\lambda .1]\)

A seismic wave is described by the equation $y(x, t)=(7.00 \mathrm{cm}) \cos [(6.00 \pi \mathrm{rad} / \mathrm{cm}) x+(20.0 \pi \mathrm{rad} / \mathrm{s}) t]\( The wave travels through a uniform medium in the \)x$ -direction. (a) Is this wave moving right ( \(+x\) -direction) or left (-x-direction)? (b) How far from their equilibrium positions do the particles in the medium move? (c) What is the frequency of this wave? (d) What is the wavelength of this wave? (e) What is the wave speed? (f) Describe the motion of a particle that is at \(y=7.00 \mathrm{cm}\) and \(x=0\) when \(t=0 .\) (g) Is this wave transverse or longitudinal?

Two speakers spaced a distance \(1.5 \mathrm{m}\) apart emit coherent sound waves at a frequency of \(680 \mathrm{Hz}\) in all directions. The waves start out in phase with each other. A listener walks in a circle of radius greater than one meter centered on the midpoint of the two speakers. At how many points does the listener observe destructive interference? The listener and the speakers are all in the same horizontal plane and the speed of sound is \(340 \mathrm{m} / \mathrm{s} .\) [Hint: Start with a diagram; then determine the maximum path difference between the two waves at points on the circle. I Experiments like this must be done in a special room so that reflections are negligible.