“The reason a lot of people do not recognize opportunity is because it usually goes around wearing overalls looking like hard work”

– Thomas A Edison

AP Physics B aspirants are expected to have a clear understanding of the standing wave modes for stretched strings fixed at both ends. They should also have a clear understanding of standing sound waves in pipes with either closed or open ends. A pipe closed at both ends is of no use and therefore of no interest to us. You should note that a

*closed*pipe means a pipe*closed at one end*. An*open*pipe means a pipe*open at both ends*.
The following multiple choice practice questions are meant for checking your understanding basic points in respect of waves and the physics of standing waves (stationary waves) in stretched strings and air columns (in pipes).

(1) Sound does not pass through

(a) steel

(b) diamond

(c) nitrogen

(d) water

(e) vacuum

Sound requires a material medium for its propagation. So sound does not pass through vacuum.

(2) When the amplitude of a wave is increased by 50%, its intensity will be

*increased by*
(a) 50%

(b) 100%

(c) 125%

(d) 150%

(e) 200%

Intensity of any wave is directly proportional to the

*square*of the amplitude. Therefore, when the amplitude becomes 1.5 times (*increment*by 50%) the original value, the intensity becomes 2.25 times (1.5^{2}times) the original intensity. The*increment*in intensity is 125% [Option (c)].
(3) Ultrasonic waves from a sonar undergoes refraction at the interface between water and air. Which one of the following characteristics of the wave remains unchanged?

(a) Wave length

(b) Speed

(c) Period

(d) Energy

(e) None of the above

The correct option is (c). The period (and of course frequency) of the wave remains unchanged.

(4) A stationary sound wave is produced in a resonance column apparatus using an electrically excited tuning fork. If P and Q are consecutive nodes, which one of the following statements is correct?

(a) If P is a position of condensation, Q is a position of rarefaction

(b) If P is a position of condensation, Q also is a position of condensation

(c) If P is a position of condensation, Q is a position of normal density (of air)

(d) Both P and Q are positions of normal density (of air)

(e) Both P and Q are positions of rarefaction

In a stationary wave the particles of the medium at the nodes will be always at rest. The phase of vibration of particles (of the medium) lying on one side of a node is opposite to the phase of vibration of particles lying on the opposite side. Therefore, if one node is a position of condensation, the next node is a position of rarefaction [Option (a)].

[Note that the particles at the antinodes will vibrate with maximum amplitude; but the air at the anti-node will have normal density (neither condensed nor rarefied]

(5) A cylindrical pipe open at both ends has a fundamental frequency

*f*in air. The pipe is dipped vertically in water so that half of its length is in water. The fundamental frequency of air column in this condition is
(a) 4

*f*
(b) 3

*f*
(d)

*f*
(e)

*f*/2
In the fundamental mode there is a node at the middle of the open pipe and the anti-nodes are at the ends. When half of the pipe is dipped in water, there is a node at the water surface and in the fundamental mode the neighbouring anti node is at the open end, out side water (fig.). The distance from node to the neighbouring anti-node is λ/4 where λ is the wave length of sound. Evidently λ/4 = half the length of the pipe so that the wave length in the fundamental mode is the same in both cases. Therefore, the fundamental frequency is

*unchanged*on dipping half the length of the pipe in water [Option (d)].
(6) A stationary wave of frequency 30 Hz is set up in a string of length 1.5 m fixed at both ends. The string vibrates with 3 segments as shown in the adjoining figure. The speed of the wave along the string is

(a) 10 ms

^{–1}
(b) 20 ms

^{–1}
(c) 30 ms

^{–1}
(d) 60 ms

^{–1}
(e) 90 ms

^{–1}
The distance between consecutive nodes (or anti-nodes) in a stationary wave is λ/2 where λ is the wave length. Therefore we have (from the figure) λ/2 = 0.5 m so that λ = 1m.

Since speed

*v*=*n λ*where*n*is the frequency we have*v =*30×1 = 30 ms

^{–1}

(7) What is the fundamental frequency of vibration of the string in the above question?

(a) 5 Hz

(b) 10 Hz

(c) 15 Hz

(d) 30 Hz

(e) 60 Hz

The speed of waves in the string is unchanged since the tension is unchanged. Since speed

*v*=*n*_{1}*λ*_{1}where*n*_{1}is the fundamental frequency and*λ*_{1 }is the wave length in the fundamental mode of vibration, we have*n*

_{1}=

*v*/

*λ*

_{1}

In the fundamental mode of vibration, the entire length of the string forms a single segment (with anti-node at the middle and nodes at the ends). Therefore we have

*λ*

_{1}/2 = length of string = 1.5 m so that

*λ*

_{1}= 3 m.

Substituting,

*n*_{1}=*v*/*λ*_{1}= 30/3 = 10 Hz.
[You can work out this problem in no time remembering that the fundamental frequency is

*one third*of the frequency with which the string vibrates with*three*segments. If the string were originally vibrating with*four*segments, the fundamental frequency would be*one fourth*].
Now see these multiple choice questions.

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