Your answers are due to be handed in by 5 pm on Wednesday, February 24 th.
1) Show by direct substitution that the following functions satisfy the wave equation.
a) y(x,t) = A(x-vt)4
b) y(x,t) = A(kx-wt)3
2) Wavelengths:
a) Radio waves travel at a speed of c=3x108m/s. FM radio waves range in frequency from 88 MHz to 108 MHz. What range in wavelengths does this correspond to? (For fun compare this to the size of your stereo’s FM antenna.)
b) Sound waves travel at a speed of around 340 m/s. We can hear sounds from as low as 20 Hz and as high as 20KHz. What range in wavelengths does this correspond to? (For fun compare this in size to a typical musical instrument or organ pipe.)
3) A transverse traveling wave on a cord is represented by the relation y(t)=6.0Cos(0.02πx+4πt) where y(t) and x are in meters and t is in seconds. For this wave determine
a) The wavelength
b) The frequency
c) The amplitude
d) The velocity of the wave (magnitude and direction)
e) The maximum speed (magnitude of velocity) of the particles of the cord.
f) The transverse displacement at x=3.5m at time t=0.26 s.
4) A wave is represented by y(t) = Acos(kx-ωt). Consider the phase of the wave.
a) Use a point of constant phase (like a peak or a trough) to show that v=dx/dt=ω/k
b) Use the distance between two peak at an instant in time to show that the wavelength λ=2π/k.
c) Use the time between two peaks at one position in x to show that the period T=2π/ω.
5) For a given tension, which travel faster: waves on a heavy string or waves on a light string? Please explain your answer both in terms of the math and in terms of what is physically happening.
6) A string resonates in four loops at a frequency of 294 Hz.
a) Name the three lower frequencies at which it will resonate.
b) Draw what the string will look like at several different times to show its motion when it is in the 4 loop mode.
7) A stretched string has resonant frequencies at 420 Hz and 315 Hz with no resonant frequencies in-between. What is the lowest resonant frequency?
8) A standing wave on a wave tray open at both ends (at x=0 and x=L) obeys the boundary conditions ∂y/∂x=0 at x=0 and at x=L. Show that the wave can be described by Y(x,t)=cos(kx)cos(ωt) and kind the values of k and ω that are allowed. Compare the frequencies that you find with the standing wave frequencies on a string of the same length but with its fixed.
9) To find the acceleration of a glider moving down an air-track, I measure its velocities
(v1 and v2) at two points and the time t that it takes to pass between them. I get the following data:
v1 = 0.21 ± 0.05 m/s, v2
= 0.85 ± 0.05 m/s and t = 8.0 ± 0.1 s.
a) Assuming that all uncertainties are independent and random, what should I report for the acceleration, a = (v2 - v1)/t and its undertainty?
b) Does my measurement agree with a theoretical prediction of 0.13 ± 0.01 m/s2?
c) If, instead of measuring the time between the two points I measure the distance to be
d = 3.740 ± 0.002 m
and then calculate the acceleration as a = (v22 - v12)/2d, what shouldbe my answer with its uncertainty?
d) Again, discuss whether my measurement and my theory agree.
10) For each of the functions below calculate ∂f/∂x, ∂f/∂y, ∂2f/∂x2, ∂2f/∂x2, and ∂2f/∂x∂y.
a) f(x,y) = sin(x - 2y)
b) f(x,y) = a2 + xy + y2