5. Interference Interference – combination of waves (an interaction of two or more waves arriving at the same place) Principle of superposition: No shift or shift by Shift by Waves source (a) If the interfering waves add up so that they reinforce each other, the total wave is larger; this is called “constructive interference”. (b) If the interfering waves add up so that they cancel each other, the total wave is smaller (or even zero); this is called “destructive interference”.

Review question: Two speakers S1 and S2 are driven by the same signal generator and are different distances from a microphone P as shown. What is the minimum frequency for constructive interference to occur at the point P? (For speed of sound use v = 340 m/s.) A. 100 B. 200 C. 400 D. 800 Review question: Two speakers S1 and S2 are separated by 2.0 m and are driven by the same signal generator. Each speaker radiates sound waves isotropically with wavelength  = 1.5 m. The sound waves from one speaker do not reflect off the other speaker. What is the number of standing wave antinodes on the line between the two speakers is? /2 = 0.75 m 1 2 3 5

6. Standing waves on a string Vibrating string can be thought of as the limit of the mass-spring system when the number of masses becomes very large. Standing waves on a string can be thought of as superposition of two traveling waves moving in opposite direction.

Wave interference, boundaries, and superposition Waves in motion from one boundary (the source) to another boundary (the endpoint) will travel and reflect.

As wave pulses travel, reflect, travel back, and repeat the whole cycle again, waves in phase will add and waves out of phase will cancel.

Standing waves on a string Different boundary conditions: Both ends fixed (see above) Both ends free (similar to both ends fixed) One end fixed and on end free (next slide)

Standing waves on a string One end fixed and on end free n=1 n=3

Question: A stretched string between 2 fixed ends has: length L = 1 Question: A stretched string between 2 fixed ends has: length L = 1.0 m, and wave speed v = 100 m/s. What is the fourth harmonic frequency of vibration of the string? A. 50 Hz B. 2100 Hz C. 150 Hz D. 200 Hz Question: A stretched string has one free end and one fixed end, and is vibrating at its 5th harmonic frequency. The number of nodes is ___. A. 1 B. 2 C. 3 D. 4 E. 5

7. Standing waves in tubes (longitudinal) Waves in tubes (pipes) can be described in terms of: displacement vibrations of the fluid pressure variations in the fluid A pressure node is a displacement antinode and vice versa Open and both ends closed pipes Closed: displacement pressure Open: pressure displacement n=1 n=2 n=3

One end open pipes (stopped pipes) displacement pressure n=1 n=3 Example: Standing sound waves are produced in a 0.6 m long pipe that is closed at the left end and open at the right end. For the first overtone, determine the locations along the pipe (measured from the left end) of the displacement nodes. Nodes at 0.0 m and 0.4 m

8. Principle of superposition and standing waves (optional)

Real pipes The pressure does not drop to zero right at the open end of a pipe. Because of this, the acoustic length is slightly grater then physical length. For a cylindrical pipe of radius r the end correction (additional length) is 0.61r One open end: Two open ends: For long pipes ( L>>r) the end correction can be neglected. For short pipes the end correction is important.

One open end: Two open ends:

E – Young’s elastic modulus ρ – density Helmholtz resonator German von Helmholtz (1821-1894) l a v – speed of sound a – area of the neck l – length of the neck V – volume V A brass, spherical Helmholtz resonator based on Helmholtz's original design, from around 1890-1900. Singing rods L – length E – Young’s elastic modulus ρ – density Sympathetic vibration: soundboards