PHYS 251 TEST #5 07/06/06 Dr. Holmes NAME
DO ALL EIGHT PROBLEMS. THE WORTH OF EACH PROBLEM IS MARKED BESIDE THE PROBLEM. SHOW YOUR WORK FOR PARTIAL CREDIT.
1. a) Write down the one-dimensional wave equation for a string:
b) Where did this equation come from (that is, is it basic, or is it derived from more basic principles - and if so, what is that basic principle or principles)?
c) Write down the one-dimensional wave equation for an electric field wave:
d) Where did this equation come from (that is, is it basic, or is it derived from more basic principles - and if so, what is that basic principle or principles)?
e) Show that Ey(x,t) = Eosin(kx-wt) is a solution of the one-dimensional wave equation for Electric Field as long as w/k = v = Ö (1/eomo) :
f) Show that Ey(x,t) = Eosin(kx-wt) is a wave that moves in the positive x direction.
2.a) What is the predicted speed for an electromagnetic wave in vacuum?
3 x 108 m/s.
b) Where does this value come from? (That is, what parameters of the vacuum does it depend on?
v = Ö [1/(e omo)]
c) If the electromagnetic wave were to travel in a transparent material different that vacuum, such as glass, would the wave travel slower, the same speed, or faster?
slower.
d) What would change in part b that would lead to your answer in part c?
e o becomes e , where e = Ke o
with K > 1 (recall K = C(with dielectric) / C(with vacuum).
3. Given that a one-dimensional wave can be described as a sine wave of amplitude 3 cm, wavelength of 35 cm, with a phase velocity of 140 m/s moving in the +x direction:
a) What is the frequency of this wave?
400 Hz.
b) Write a function, y(x,t), for this wave:
(be sure to have only x and t show up as variables; all other quantities should have values with units included)
y(x,t) = 3 cm * sin[ (17.95 rad/m)*x - (2513 rad/sec)*t]
c) If the wavelength and power did NOT change but the physical medium did change, would the following quantities change? Answer each with one of the following: [ Yes: would change; No: would not change; Maybe: might change depending on whether something else changed]
amplitude: yes, would change
wavelength: yes, would change
phase velocity: yes, would change
4. a) DESIGN a string (choose its length, mass density, and tension) that when plucked will have a fundamental frequency of 250 Hz.
b) Based on your design, what is the phase velocity of waves on this string?
5. a) Given that the intensity of a sound wave of frequency 2,800 Hz is 3.5 x 10-4 W/m², what is the intensity in dB?
85.44 dB
b) Given that the intensity of a sound wave of frequency 750 Hz is 73 dB, what is the intensity in Watts/m² ?
2.0 x 10-5 W/m2 .
c) Does the frequency enter into calculations for part a only, for part b only, for both, or for neither?
neither.
d) If two waves with the two frequencies above (2,800 Hz and 750 Hz) had the same amplitude, would the larger frequency have a larger, the same, or smaller speed (phase velocity)?
same .
6. a) Given that the intensity of a sound wave of frequency 2,800 Hz is 3.5 x 10-4 W/m² (same as in 5a above), what will its intensity be if the amplitude of the sound wave is increased by a factor of 3:
in Watts/m²: 3.15 x 10-3 W/m2
in dB: 95 dB
b) Given that the intensity of a sound wave of frequency 750 Hz is 73 dB (same as in 5b above), what will its intensity be if the amplitude stays the same but the frequency is decreased by a factor of 1/10:
in Watts/m²: 2.0 x 10-7 W/m2 .
in dB: 53 dB
c) Given that the intensity of a sound wave of frequency 2,800 Hz is 3.5x10-4 W/m2 (same as in 5a above) at a distance of 3 meters from the speaker, what will its intensity be at a distance of 15 meters from the speaker:
in W/m²: 1.4. x 10-5 W/m2 .
in dB: 71.5 dB
7. a) What is the speed of sound in air in a freezer if the temperature of the freezer is -30 oF ?
309.5 m/s.
b) What are the nominal lower and upper limits on the frequency of SOUND that are audible to the human ear in the above freezer?
lower: 20 Hz.
upper: 20,000 Hz.
c) What is the wavelength of the SOUND (in air) in the above freezer?
for the lower frequency: 15.47 meters.
for the upper frequency: 15.47 mm.
d) An FM radio station broadcasts its signal over an electromagnetic wave with a frequency of 91.1 MHz. What is the wavelength for this electromagnetic wave?
3.29 meters.
e) An AM radio station broadcasts its signal over an electromagnetic wave with a frequency of 1020 kHz. Is the wavelength for this wave longer, the same, or shorter than for the FM radio station of part d?
longer.
8. Consider a speaker that produces a sound of frequency 2,800 Hz. (In parts a and b, consider the air to be still - no wind.)
a) Suppose that the speaker is held stationary, and you head away from the speaker at a speed of 32 m/s. What frequency will you measure for the sound?
2,540.3 Hz.
b) Suppose that you are stationary and the speaker moves towards you at a speed of 32 m/s. What frequency will you now measure for the sound?
3,086.3 Hz.
c) Suppose that the speaker of frequency 2,800 Hz is mounted on a train that is leaving the station going East at 20 m/s, you are approaching the station and the back of the train going East at a speed of 32 m/s, and there is a West wind (blowing towards the East from the West) at a speed of 12 m/s (relative to the station). What will you measure for the frequency of the sound [you must be precise: answer to the closest Hz]?
2,895.2 Hz.