Various Practice Problems

Physics questions

The expression given below represents a transverse harmonic wave travelling along a
string which is stretched along the horizontal x-axis. The expression yields the transverse
wave displacement y in metres when x is entered in metres and the time t in seconds.
For 1 mark each, answer the following questions (be sure to give physical units with your
answers):
[2] (a) What are the frequency and wavelength of this wave?
[1] (b) What is the propagation speed c of this wave?
[1] (c) In what direction is this wave travelling?
[2] (d) What is the maximum transverse velocity achieved by any particle of the string?
?
y(x,t) = .05 sin(60t + 2x +1)
Phys 321 Module 10 Exam your ID
Fall, 2002 page 3 of 10
continued.....
A2. On a winter day at a location near Calgary, a
layer of cold Arctic air at -40°C occupies a
valley bottom. Above the cold air is a layer of
warmer Pacific air at +20°C. To a good
approximation, the air pressure in both layers
can be assumed to be the same and equal to
8.00x104 Pa.
[3] (a) Find the numerical value of the ratio of acoustic impedances of the two layers. Which
has the higher impedance? (Recall gas density is inversely proportional to temperature at
constant pressure.)
[1] (b) A sound wave is beamed vertically upward towards the horizontal interface between
the cold air and the warmer air. Is the reflection coefficient at the interface positive
or negative? Explain very briefly.
[2] (c) What fraction of the energy carried by the sound wave of part (b) is reflected from the
interface?
-40° C
+20° C
Phys 321 Module 10 Exam your ID
Fall, 2002 page 4 of 10
continued.....
A3. An isotropic point source of sound waves produces an intensity level of 83.0 dB at a
distance of 1.00!m from the source. The source is surrounded by a uniform medium
consisting of air at 20°C with sound speed 344 m/s.
[2] (a) What is the intensity level (in dB) 1000 m from the source, assuming absorption of
0.01 dB/m in the surrounding air?
[2] (b) What is the acoustic intensity (in W/m2) at a distance of 1000 m from the source?
[2] (c) Calculate the total power being emitted by the source at any instant.
(Iref for the decibel scale is 1x10-12 W/m2.)
Phys 321 Module 10 Exam your ID
Fall, 2002 page 5 of 10
continued.....
A4. The diagram shows a plot of a triangular transverse wave pulse as a function of x at t=0.
Note that the vertical scale on the plot is different from the horizontal scale. The pulse is
moving to the right on a string under tension 600 N at wave speed c=40 m/s.
Using the axes provided, plot the transverse particle velocity vy and the transverse force
Fy for this pulse as functions of x at t=0. Be careful to label your vertical axes with
appropriate scales. Scales should include physical units.
[3]
[3]
y(x,0)
vy(x,0)
P(x,0)
x
x
x
-2 m -1 m 0 +1 m
-2 m -1 m 0 +1 m
-2 m -1 m 0 +1 m
5 cm
Fy(x,0)
40 m/s
Phys 321 Module 10 Exam your ID
Fall, 2002 page 6 of 10
continued.....
A5. A liquid has bulk modulus 9.1x108 Pa and density 0.81x103 kg/m3. A plane harmonic
acoustic travelling wave of frequency 1000 Hz creates an acoustic intensity of 1.00x10-6 W/m2
at a point P in the liquid.
[1.5] (a) What is the speed of acoustic waves in this liquid?
[1.5] (b) What is the acoustic impedance of this liquid?
(c) At an instant of time when the velocity of the fluid element at P is zero, what are
[1.5] (i) the pressure fluctuation at P?
[1.5] (ii) the absolute value of the displacement of the fluid element at P?
Phys 321 Module 10 Exam your ID
Fall, 2002 page 7 of 10
continued.....
A6. A guitar string has mass per unit length 3.00 grams/metre and is stretched under a
tension of 1500N with its ends fixed a distance 80.0 cm apart.
[2] (a) What is the frequency of the fundamental mode of vibration (i.e., the 1st harmonic) of the
string under these conditions?
[2] (b) Suppose the x-axis below lies along the guitar string, with the fixed ends at x=0 and
x=80 cm. Sketch the shape of the displacement y of the string when it is vibrating in its
second overtone (i.e., 3rd harmonic) under the given tension. Assume the particle at
x=20!cm has a negative displacement, as indicated.
[2] (c) Referring to the situation you drew in part (b), suppose that the displacement of the
particle at x=20 cm is -0.800 mm at the instant depicted in the figure. What then is the
maximum displacement to be found anywhere along the guitar string at this same instant?
x
y
0 20 cm 40 cm 60 cm 80 cm
particle at x=20 cm
fixed end fixed end
Phys 321 Module 10 Exam your ID
Fall, 2002 page 8 of 10
continued.....
y(x,0)
x
1 m/s
-1 m 0 m !2 m
y = 1 m
Part B. [9 marks] Attempt ONE of the two problems below. Use the
following two blank pages for your answer.
B7. The transverse wave pulse shown below travels on a stretched string in the positive xdirection
at wave propagation speed 1 m/s. The tension in the string is 1 N. The
displacement y of the pulse at t=0 is given by the function to the right of the figure. This
function returns the value of y in metres when x is entered in metres.
[5] (a) Find the particle velocity at x="2 m as a function of time, i.e., find vy("2,t). Draw a
reasonably accurate plot of vy("2,t) versus t.
[4] (b) Find the total energy (in joules) being transported by this wave pulse.
OR
B8. Two loudspeakers, A and B, are positioned on the x-axis at
x = 0 m and x = +1.5!m. The loudspeakers act as isotropic
point sources, operating in phase and emitting pure tones of
frequency 340 Hz. Each loudspeaker emits a power of
10.0!W uniformly in all directions. The speed of sound in
air is 340!m/s, and absorption in the air may be neglected.
[5] (a) Find the resultant acoustic intensity produced by the
loudspeakers at point P located on the x-axis at x=+20.0 m.
Note that interference effects must be considered in
calculating this intensity, and that 1/r2 attenuation (i.e. geometrical spreading of the sound
waves) should be taken into account. Assume that there is no blocking of the sound
transmission from the more distant speaker (A) by the one nearer to P.
[4] (b) If you started at point P and travelled along the circumference of the circle shown in the
diagram, what distance (arc length) would you have to travel before you encountered a
local maximum in the acoustic intensity due to the two coherent sources?
?