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:

Principle of superposition and standing waves (optional)

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

The waves spread out from the opening! Diffraction 1) What it is? The bending of waves behind obstacles or apertures into the ”shadow region”, that can be considered as interference of many waves. 2) Haw to observe? Diffraction is most pronounced when the wavelength of the wave is similar to the size of the obstacle or aperture. For example, the diffraction of sound waves is commonly observed because the wavelength of sound is similar to the size of doors, etc. The waves spread out from the opening! Waves will diffract around a single slit or obstacle. The resulting pattern of light and dark stripes on a screen is called a diffraction pattern (fringes). This pattern arises because different points along a slit create wavelets that interfere with each other just as a double slit would.

3a) Diffraction from a single slit (intensity) Minima (dark fringes): Example: In order to obtain a good single slit diffraction pattern, the slit width could be: A. /100 ; B. /10; C. ; D. 10; E.100

Review question: Two speakers S1 and S2 are driven by the same signal generator and are different distances from a microphone P as shown. The minimum frequency for constructive interference to occur at P is __ Hz. (The speed of sound is 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. The number of standing wave antinodes on the line between the two speakers is __. 1 2 3 5