SoundChapter 17

Wave fronts

In a flat region of space, spherical wave fronts are near planar

• Longitudinal waves

consist of zones of high and low frequency

OctaveMusical notes that differ by a factor of 2 in frequency are separated by an octave

UltrasoundSound waves above normal human hearing range (>20000Hz)

Intensity of Sound I

= P/A

Intensity is expressed on a logarithmic scale, (dB) decibel

Intensity level in dB = (10dB) x log10)

Example• Express the threshhold of hearing (2.5 x 10-12 W/m2) and the

threshhold of pain (1.0 W/m2) in decibels.

Solution:(10dB)x log10 (2.5x 10-12 W/m2) = (10dB) x log10 (2.5) = 4.0 dB

1.0x 10-12 W/m2

(10dB)x log10 (1.0 W/m2) = (10dB) x log10 1012) = 120 dB

1.0x 10-12 W/m2

• As a sound wave spreads out from it source, its intensity falls off because the area of the wavelength grows larger;

the wave energy per unit area grows smaller

andI 1∝1/4𝜋 r 12 I 2∝1/4 𝜋 r 22

(10dB)x log10 (0.28W/m2) = (10dB) x log10 (0.28 x 1012) = 114 dB 1.0x 10-12 W/m2

At a distance of 30m from a jet engine, the intensity of sound is 10W/m 2 , and the intensity level of 130dB. What are the intensity and intensity level at a distance of 180dB?

I 2 = r12 I1 = (30m)2 I1 = (2.8 x 10-2) x I1 = (2.8 x 10-2) x (10W/m2) = 2.8 W/m2

r22 (180m)2

The Speed of Sound : Standing Waves• Speed of sound (in air, 0⁰C, 1 atm) = 331 m/s

The standing sound wave in the column of air in a tube closed at one end must have a displacement node at the closed end and and antinode at the open end

Only odd multiples are possible λ1 = 4L, λ2 = 4/3L, λ3 = 4/5L, λ4 = 7/4L, …

Eigenfrequencies: (f = v/ λ)

f1 = v/4L, f2 = 3v/4L, f3 = 5v/4L, …

A tube open at both ends:

eigenfrequencies

The Doppler Shifthttp://www.astro.sunysb.edu/mzingale/software/astro/doppler.avi

A receiver will detect a higher frequency when the source is approaching, and a lower frequency when the source is moving away from the receiver.

f’/ f = v’/v f’ = f (1 ± VR/v)

Doppler shift, moving receiver

Example:

Suppose that a stationary siren emits a tone of frequency 440 Hz as the train moves away from it at 30.0m/s. What is the frequency received on the train?

A motorboat speeding at 6.0 m/s is moving in the same direction as a group of water waves of frequency 0.62 Hz and speed 2.5 m/s (relative to the water). What is the frequency with which the wave crests pound on the motorboat?

f’ = f (1 - VR/v) f’ = 440Hz (1 – 30 m/s/331 m/s)= 400Hz

f’ = f (1 - VR/v) f’ = 0.62Hz (1 – 6.0m/s/2.5 m/s)= -0.87Hz

Doppler shift, moving emitter

f’

Example:Suppose that a stationary siren emits a tone of frequency 440 Hz as the train recedes from a stationary observer at 30.0m/s. What frequency does the observer hear?

f’ = ] = 403 Hz

When the speed of an aircraft exceeds the speed of sound, the aircraft overtakes the wave fronts. The sound is confined to a conical region, the Mach cone

In a time, t, the aircraft moves a distance VE t and the initial sound wave moves vt.

sin 𝜃=v t /V E

The sharp pressure disturbance at the surface of the cone is heard as a loud bang = sonic boom

Diffraction• Deflection of waves at the edge of an obstacle• The amount of diffraction increases with the ratio of

wavelength to the size of the gapfanlike beamsspread outforming adiffraction pattern

Example:The minimum size of a structure that can be detected in an ultrasound image is limited by diffraction. With careful analysis of echoes reflected from structures within the body, features as small as one-quarter of a wavelength can be imaged. Suppose the probe uses ultrasound with a frequency of 2.0 x 106Hz. What is the smallest feature that can be detected?

λ