Speed of Waves ρ - density ρ - density Speed of waves in strings (review): F - tension m - mass of the string L - length of the string µ - linear density Speed of longitudinal waves in a solid rod: E - Young’s modulus m - mass V - volume ρ - density Speed of sound waves in a fluid: We studied strings before, and had a lab. Solid rod, fluid and gas are new for them. Equations are similar, and this is enough. B - bulk modulus ρ - density

Speed of sound waves in an ideal gas: T - temperature in Kelvin t - temperature in Celsius a - constant For air: Most students do not know what it is “heat capacity” , “molar mas”, etc. For them these are some constants, and this is enough. However the last equation is important for them because it was used in the recent lab.

Young’s and bulk modulus (optional) The Young’s modulus is the stress (pressure) divided by the strain (relative change of length) due to the pressure The bulk modulus is the pressure divided by the relative volume change due to the pressure P – pressure V - volume More about sound waves in an ideal gas (optional) γ - ratio of heat capacities M - molar mass R - universal gas constant; R = 8.31451 J/(mol⸱K) For air:

Speed of Sound in Some Common Substances Substance Speed (m/s) 1. Air (20 oC) 344 2. Helium (20 oC) 999 3. Water (0 oC) 1,402 4. Lead 1,200 5. Human tissue 1,540 6. Aluminum 6,420 7. Iron and steel 5,941

Reflection of waves a) Mirror reflection in out S S’ source image normal to surface S S’ source image wall (mirror) b) Diffuse reflection It is easy to demonstrate reflection and refraction of light, and it is more difficult for sound. However, everyone knows echo. c) Reflection from curved surfaces acoustical mirrors (curved mirrors) whispering galleries

Refraction: Snell’s Law In optics: index of refraction: Example: n1 n2 n3 Sound waves in atmosphere when temperature varies with height Sound traveling against the wind Mirage

Doppler Effect What do you hear when a sound-emitting object (train, car) passes you? The sound changes in pitch (frequency) as the object goes. Why does this happen? As the object travels towards you, the distance between wavefronts is compressed; this makes it seem like  is smaller and frequency is higher. As the object travels away from you, the distance between wavefronts is extended; this makes it seem like  is larger and frequency is lower. Because the speed of a wave in a medium is constant, change in  will affect change in f. Velocity of listener (L): vL Velocity of source (S): vS Velocity of sound: v There is nice demo vL or vS is “+” if in the same direction as from listener to source and is “-” otherwise

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

Example: Light of wavelength 610 nm is incident on a slit 0 Example: Light of wavelength 610 nm is incident on a slit 0.20 mm wide and the diffraction pattern is produced on a screen that is 1.5 m from the slit. What is the width of the central maximum? Example: Light of wavelength 687 nm is incident on a single slit 0.75 mm wide. At what distance from the slit should a screen be placed if the second dark fringe in the diffraction pattern is to be 1.7 mm from the center of the screen?