Chapter 17 Sound Waves: part one

Introduction to Sound Waves Sound waves are longitudinal waves They travel through any material medium The speed of the wave depends on the properties of the medium The mathematical description of sinusoidal sound waves is very similar to sinusoidal waves on a string

Categories of Sound Waves The categories cover different frequency ranges Audible waves are within the sensitivity of the human ear Range is approximately 20 Hz to 20 kHz Infrasonic waves have frequencies below the audible range Ultrasonic waves have frequencies above the audible range

Speed of Sound Waves Use a compressible gas as an example with a setup as shown at right Before the piston is moved, the gas has uniform density When the piston is suddenly moved to the right, the gas just in front of it is compressed Darker region in the diagram

Speed of Sound Waves, cont When the piston comes to rest, the compression region of the gas continues to move This corresponds to a longitudinal pulse traveling through the tube with speed v The speed of the piston is not the same as the speed of the wave

Speed of Sound Waves, General The speed of sound waves in a medium depends on the compressibility and the density of the medium The compressibility can sometimes be expressed in terms of the elastic modulus of the material The speed of all mechanical waves follows a general form:

Speed of Sound in Liquid or Gas The bulk modulus of the material is B The density of the material is  The speed of sound in that medium is

Speed of Sound in a Solid Rod The Young’s modulus of the material is Y The density of the material is  The speed of sound in the rod is

Speed of Sound in Air The speed of sound also depends on the temperature of the medium This is particularly important with gases For air, the relationship between the speed and temperature is The 331 m/s is the speed at 0 o C T C is the air temperature in Celsius

Speed of Sound in Gases, Example Values Note temperatures, speeds are in m/s

Speed of Sound in Liquids, Example Values Speeds are in m/s

Periodic Sound Waves A compression moves through a material as a pulse, continuously compressing the material just in front of it The areas of compression alternate with areas of lower pressure and density called rarefactions These two regions move with the speed equal to the speed of sound in the medium

Periodic Sound Waves, Example A longitudinal wave is propagating through a gas- filled tube The source of the wave is an oscillating piston The distance between two successive compressions (or rarefactions) is the wavelength Use the active figure to vary the frequency of the piston

Periodic Sound Waves, cont As the regions travel through the tube, any small element of the medium moves with simple harmonic motion parallel to the direction of the wave The harmonic position function is s (x, t) = s max cos (kx –  t) s max is the maximum position from the equilibrium position This is also called the displacement amplitude of the wave

Periodic Sound Waves, Pressure The variation in gas pressure,  P, is also periodic  P =  P max sin (kx –  t)  P max is the pressure amplitude It is also given by  P max =  v  s max k is the wave number (in both equations)  is the angular frequency (in both equations) s (x, t) = s max cos (kx –  t)

17.4 Traveling Sound Waves

Periodic Sound Waves, final A sound wave may be considered either a displacement wave or a pressure wave The pressure wave is 90 o out of phase with the displacement wave The pressure is a maximum when the displacement is zero, etc.

Example, Pressure and Displacement Amplitudes

17.5: Interference

Phase difference  can be related to path length difference  L, by noting that a phase difference of 2  rad corresponds to one wavelength. Therefore, Fully constructive interference occurs when  is zero, 2 , or any integer multiple of 2 . Fully destructive interference occurs when  is an odd multiple of  :

Standing Waves The resultant wave will be y = (2A sin kx) cos  t This is the wave function of a standing wave There is no kx –  t term, and therefore it is not a traveling wave In observing a standing wave, there is no sense of motion in the direction of propagation of either of the original waves

Standing Waves in Air Columns Standing waves can be set up in air columns as the result of interference between longitudinal sound waves traveling in opposite directions The phase relationship between the incident and reflected waves depends upon whether the end of the pipe is opened or closed Waves under boundary conditions model can be applied

Standing Waves in Air Columns, Closed End A closed end of a pipe is a displacement node in the standing wave The rigid barrier at this end will not allow longitudinal motion in the air The closed end corresponds with a pressure antinode It is a point of maximum pressure variations The pressure wave is 90 o out of phase with the displacement wave

Standing Waves in Air Columns, Open End The open end of a pipe is a displacement antinode in the standing wave As the compression region of the wave exits the open end of the pipe, the constraint of the pipe is removed and the compressed air is free to expand into the atmosphere The open end corresponds with a pressure node It is a point of no pressure variation

Standing Waves in an Open Tube Both ends are displacement antinodes The fundamental frequency is v/2L This corresponds to the first diagram The higher harmonics are ƒ n = nƒ 1 = n (v/2L) where n = 1, 2, 3, …

Standing Waves in a Tube Closed at One End The closed end is a displacement node The open end is a displacement antinode The fundamental corresponds to ¼ The frequencies are ƒ n = nƒ = n (v/4L) where n = 1, 3, 5, …

More About Instruments Musical instruments based on air columns are generally excited by resonance The air column is presented with a sound wave rich in many frequencies The sound is provided by: A vibrating reed in woodwinds Vibrations of the player’s lips in brasses Blowing over the edge of the mouthpiece in a flute

Resonance in Air Columns, Example A tuning fork is placed near the top of the tube When L corresponds to a resonance frequency of the pipe, the sound is louder The water acts as a closed end of a tube The wavelengths can be calculated from the lengths where resonance occurs

Spatial and Temporal Interference Spatial interference occurs when the amplitude of the oscillation in a medium varies with the position in space of the element This is the type of interference discussed so far Temporal interference occurs when waves are periodically in and out of phase There is a temporal alternation between constructive and destructive interference

Beats Temporal interference will occur when the interfering waves have slightly different frequencies Beating is the periodic variation in amplitude at a given point due to the superposition of two waves having slightly different frequencies

Beat Frequency The number of amplitude maxima one hears per second is the beat frequency It equals the difference between the frequencies of the two sources The human ear can detect a beat frequency up to about 20 beats/sec

Beats, Final The amplitude of the resultant wave varies in time according to Therefore, the intensity also varies in time The beat frequency is ƒ beat = |ƒ 1 – ƒ 2 |