• Main points of last lecture:• Sound intensity:

– I=P/A, I0=1x10-12 W/m2

• Spherical waves

• Dopper shift:

• Main points of today’s lecture:• Interference of sound waves

• Standing waves on string:

• Standing wave in air columns:– both ends open

– one end open

• Beat Phenomena:

Physics 231 Lecture 36

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Final Exam

• A common final exam time is scheduled for all sections of Physics 231 • Time: 8:00-10:00 PM, Tues., Dec. 12• Each section has a different exam location.• Location for Section 1: Wells Hall B106

• An alternate exam time has been scheduled for students who have conflicts with the regular time. – Six students have confirmed conflicts with me and will take the

exam then.

– You must have permission from me to take the exam at the alternate time.

• Alternate time: 7:45-9:45 AM, Monday, Dec. 11• Location: BPS 1415 (this room)

Example

• A train at rest emits a sound at a frequency of 1000 Hz. An observer in a car travels away from the sound source at a speed of 30.0 m/s. What is the frequency heard by the observer? (assume vs=343 m/s)– a) 513 Hz– b) 713 Hz– c) 913 Hz– d) 1013 Hz– e) 1113 Hz

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Conceptual quiz• Three observers,A,B, and C are listening to a moving source of sound.

The diagram below shows the location of the wavecrests of the moving source with respect to the three observers.Which of the following is true?– a. The wavefronts move faster at A than at B and C.– b. The wavefronts move faster at C than at A and B.– c. The frequency of the sound is highest at A.– d. The frequency of the sound is highest at B.– e. The frequency of the sound is highest at C.

Interference of sound waves

• Assume both speakers are connected to the same signal generator.Then these sound waves interfere as follows:– Constructive interference occurs when the path difference between

two waves’ motion is zero or some integer multiple of wavelengths• path difference = | r1-r2 | = nλ

– Destructive interference occurs when the path difference betweentwo waves’ motion is an odd half wavelength

• path difference = | r1-r2 | = (n + ½)λ

Fig 14.15, p. 441

Slide 17

Interference example• If both speakers are driven with the signal generator, what kind of

interference will be along the center line passing through O– a) Constructive– b) Destructive

• If point P is the first interference minimum as one moves away from O, | r1-r2 | =– a) 0– b) λ/2– c) 3λ/2– d) λ– e) 2 λ

Standing waves on a string

• Two traveling sinusoidal waves of the same frequency moving in opposite directions can superimposed so as to make a standing wave:

• Because the string is held at each end, those points are “nodes” of the oscillator motion, where the amplitude of motion vanishes.

• Anti-nodes are the points where the motion is maximal. Fig 14.18, p. 443

Slide 25

• Motion with one antinode has λ/2 = L, two antinodes has λ=L...

• The general relationship is:

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“fundamental” “harmonics”

Example

• A stretched string fixed at each end has a mass of 40.0 g and a length of 8.00 m. The tension in the string is 49.0 N. (a) Determine the positions of the nodes and antinodes for the 2nd harmonic. (b) What is the vibration frequency for this harmonic?

Fig 14.18, p. 443

Slide 25

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Standing waves in an air columns: both ends open

• If one end of the air column is closed, a node must exist at this end since the movement of the air is restricted

• If the end is open, the elements of the air have complete freedom of movement and an antinode exists

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Standing waves in an air columns: one end closed

• If one end of the air column is closed, a node must exist at this end since the movement of the air is restricted

• If the end is open, the elements of the air have complete freedom of movement and an antinode exists

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Quiz

• The fundamental frequencies of two air columns are the same. Column A is open at both ends, while column B is open at only one end. The length of column A 0.60 m. What is the length of column B?– a) 0.1 m– b) 0.2 m– c) 0.3 m– d) 0.5 m– e) 0.6 m

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Conceptual quiz

• The traces below show beats that occur when two different pairs of waves are added. For which of the two is the difference in frequency of the original waves greater?– 1. Pair 1– 2. Pair 2– 3. The frequency difference was the same for both pairs of waves.– 4. Need more information.

Example

• Each of three tuning forks A, B, and C has a slightly different frequency. When A and B are sounded together, they produce a beat frequency of 2 Hz. When A and C are sounded together, the produce a beat frequency of 5 Hz. What is the beat frequency that occurs when B and C are sounded together? There are two possible answers.

7 3Hz 105or 95Hz 102or 98

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Forced vibrations and resonance

• A system with a driving force will force a vibration at its frequency• When the frequency of the driving force equals the natural frequency

of the system, the system is said to be in resonance

Conceptual question

• A string is clamped at both ends and plucked so it vibrates in astanding mode between two extreme positions a and b. Let upward motion correspond to positive velocities.When the string is in position c, the instantaneous velocity of points along the string:– a. is zero everywhere.– b. is positive everywhere.– c. is negative everywhere.– d. depends on location.

Example

• Two trucks travel at the same speed. They are far apart on adjacent lanes and approach each other essentially head-on. One driver hears the horn of the other truck at a frequency that is 1.2 times thefrequency he would hear if the trucks were stationary. The speed of sound is 343 m/s. At what speed is each truck moving?

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