1.
Questions: 1 2 3 4 5 6 7 8 9 10
Physics 1100: Waves Solutions
1. A fisherman notices that his boat is moving up and down in a periodic way because of waves on the
surface of the water. It takes 4.0 s for the boat to travel from its highest point to its lowest, a total distance
of 3.0 m. The fisherman sees that the wave crests are spaced 8.0 m apart.
(a) How fast are the waves moving?
(b) What is the amplitude, frequency, wavelength, and period of the waves?
The period, T, is the time that is takes the motion to repeat itself. The motion is peak down to trough and
back up to the peak. Since it took 4.0 s to move from peak to trough, it should also take 4.0 s to move
from trough to peak. The period is T = 8.0 s.
The frequency is given by f = 1/T = 1 / 8.0 s = 0.125 Hz.
The wavelength, λ, is the distance that over which the wave repeats itself. The peaks are 8.0 m apart, so λ
= 8.0 m.
The amplitude, A, of a wave is the height of the wave above average height, or onehalf the peakto
trough distance. Hence A = ½(3.0 m) = 1.5 m.
The speed of a wave is v = λ/T = 8.0 m / 8.0 s = 1.0 m/s.
2. Radio waves travel at the speed of light, 3.00 × 108 m/s. (a) AM radio waves have a frequency range of
530 kHz to 1600 kHz. What range of wavelengths does this correspond to? (b) FM waves have
wavelengths of 2.77 m to 3.40 m. What range of frequencies does this correspond to?
4. The speed of a wave on a string is given by the formula , where is the linear density given
by . Thus the speed is
.
If we double the tension, v = 89.1 m/s.
If we double the mass, v = 44.5 m/s.
7. The waves on a certain string move at 50.0 m/s. The string is 7.5 m long. If both ends are fixed, what are
the first five resonance or standing wave frequencies? If one end is free, what are the first five frequencies?
Is there any way of telling that a string is fixed or free at one end by looking at a sequence of resonance
frequencies?
(i) The standing wave of resonance frequencies of a string fixed at both ends is given by the formula
where n = 1,2,3,… Since we are given v and L, we have
fn = n(50 m/s)/(2 7.5 m) = n(3.33 Hz).
The first five frequencies are thus: f1 = 3.33 Hz, f2 = 6.67 Hz, f3 = 10.0 Hz, f4 = 13.33 Hz, and f5 =
16.67 Hz.
(ii) The standing wave of resonance frequencies of a string fixed at only one end is given by the formula
where n = 1,3,5, ... Since we are given v and L, we have
fn = n(50 m/s)/(4 7.5 m) = n(1.67 Hz).
The first five frequencies are thus: f1 = 1.67 Hz, f3 = 5.00 Hz, f5 = 8.33 Hz, f7 = 11.67 Hz, and f9 =
15.00 Hz.
(iii) If we have a sequence of resonance frequencies, we can tell if the string is fixed at both ends or open
at one end by looking at the ratio of the frequencies. To find the ratio divide through by the greatest
common divisor. For a string fixed at both ends, the ratio will have both even and odd numbers. A string
fixed at only one end will have only odd numbers. For example, from part (i) above, the ratio is 1:2:3:4:5,
where the common factor 3.33 Hz is factored out. In part (ii), the ratio is 1:3:5:7:9, where the common
factor 1.67 Hz is factored out. Notice that the greatest common factor is the fundamental frequency.