• The average intensity of a wave is the rate at • Threshold of hearing which the energy flows through a unit area, A, – Faintest sound most humans can hear oriented perpendicular to the direction of travel – About 1 x 10-12 W/m2 of the wave • Threshold of pain 1 ΔE P I = = – Loudest sound most humans can tolerate A Δt A – About 1 W/m2 • The rate of energy transfer is the power • The ear is a very sensitive detector of • Units are W/m2 sound waves – It can detect pressure fluctuations as small as about 3 parts in 1010
Plane Wave Spherical Waves • A spherical wave • Far away from the propagates radially source, the wave fronts outward from the are nearly parallel planes oscillating sphere • The rays are nearly • The energy propagates parallel lines equally in all directions • A small segment of the • The intensity is wave front is P P approximately a plane I = av = av wave A 4πr 2
Intensity of a Point Source Intensity Level of Sound Waves
• Since the intensity varies as 1/r2, this is an • The sensation of loudness is logarithmic in inverse square relationship the human hear • The average power is the same through any • β is the intensity level or the decibel level spherical surface centered on the source of the sound • To compare intensities at two locations, the I inverse square relationship can be used β = 10 log Io 2 Ir • I is the threshold of hearing, 1.0 x 10-12 12= o 2 W/m2 Ir21
Various Intensity Levels Decibel Scale Example
• Threshold of hearing is 0 dB Two people hear a songbird singing. One • Threshold of pain is 120 dB person, only 1.00 m from the bird, hears the sound with an intensity of 2.80 x 10-6 W/m2. • Jet airplanes are about 150 dB • What is the decibel level of the sound? • Multiplying a given intensity by 10 adds 10 • What intensity is heard by a second dB to the intensity level person standing 4.25 m from the bird • See Fig. 14-14, compare to 14-2 (assuming no reflected sound). • What is the power output of the bird’s song?
Sound vs. Light The Doppler Effect
• Sound waves and light waves are not the • Definition: “The change in wavelength of same thing! radiation due to relative radial motion • You don’t actually hear radio waves, you between the source and the observer.” hear sound waves that were picked up by • Radial: Doppler effect only works for line- the radio of-sight motion • Ears are sensitive to vibrations/ • Relative: Effect will happen whether the compressions in the air source, observer, or both are moving. • Eyes are sensitive to light waves (but only a very small portion of them…)
Doppler Effect Directions in the Doppler Effect
• v = wave speed, u = source/observer speed • Motion towards each other (decreasing distance): higher pitch • Motion away from each other (increasing distance): lower pitch
1± u v f ! = ( o ) f Demo: http://astro.unl.edu/classaction/animations/light/dopplershift.html (1 us v)
Example: A dog whistle produces Interference & Superposition sound waves with a frequency of 30 kHz. If the human range of • Two traveling waves can meet and pass through each other without being hearing begins around 20 kHz, how destroyed or even altered fast would someone have to run • Waves obey the Superposition Principle with a dog whistle for a human – If two or more traveling waves are moving through a medium, the resulting wave is found being to be able to hear it? Should by adding together the displacements of the they run towards the listener or individual waves point by point – Actually only true for waves with small away from them? amplitudes
Constructive Interference Constructive Interference in a String • Two waves, a and b, have the same frequency and amplitude – Are in phase • The combined wave, c, has the same • Two pulses are traveling in opposite directions frequency and a • The net displacement when they overlap is the greater amplitude sum of the displacements of the pulses • Note that the pulses are unchanged after the interference
Destructive Interference in a Destructive Interference String
• Two waves, a and b, have the same amplitude and frequency • They are 180° out of phase • Two pulses are traveling in opposite directions • When they combine, • The net displacement when they overlap is decreased since the displacements of the pulses the waveforms cancel subtract • Note that the pulses are unchanged after the interference
Reflection of Waves – Fixed End Reflected Wave – Free End
• Whenever a traveling wave reaches a boundary, some or all of the wave is reflected • When it is reflected from a fixed end, the wave is inverted • When a traveling wave reaches a boundary, • The shape remains the all or part of it is reflected same • When reflected from a free end, the pulse is not inverted
Interference of Sound Waves Standing Waves on a String • Constructive interference occurs when the path difference between two waves’ motion is zero or some integer multiple of wavelengths – path difference: Δd = nλ • Destructive interference occurs when the path difference between two waves’ • Nodes must occur at the ends of the string motion is an odd half wavelength because these points are fixed – path difference: Δd = (n + ½)λ
Standing Waves on a String Sound Wave Interference Example
Two speakers on opposite ends of a basketball • The lowest frequency of court (28m wide) emit sound waves in phase vibration (b) is called with a frequency of 171.5 Hz. the fundamental 1. If someone stood exactly halfway between frequency the two speakers, would they hear
• ƒ1, ƒ2, ƒ3 form a constructive or destructive interference? harmonic series 2. If the person wanted to hear destructive interference, how much closer should they m F move towards one speaker (either side)? ƒ = mƒ = n 1 2L µ
Standing Sound Waves Tube Open at Both Ends (flute)
There are three categories of standing sound waves, depending on their structure: • Open-open (both ends are antinodes) • Closed-closed (both ends are nodes) • Open-closed (one end is a node, other is an antinode).
Open-open and closed-closed tubes behave in the same way because they both support symmetrical standing waves.
Harmonics in Open-open or Tube Closed at One End (clarinet) Closed-closed tube • In a pipe open at both ends, the natural frequency of vibration forms a series whose harmonics are equal to integral multiples of the fundamental frequency • This also works for closed-closed tubes!
v ƒ = m = mƒ m = 1, 2, 3,… m 2L 1
Resonance in an Air Column Closed at One End Beats • The closed end must be a node • Beats are variations in loudness due to interference • Two waves have slightly different frequencies and • The open end is an antinode the time between constructive and destructive interference alternates v f = m = mƒ m = 1, 3, 5,… • The beat frequency equals the difference in m 4L 1 frequency between the two sources:
• There are no even multiples of the ƒ = ƒ − ƒ fundamental harmonic b 1 2
Doppler Effect and Beats: Beats Example Speeding! The second-chair saxophone player is trying to get in tune, listening to the first-chair player. The first-chair player is playing a note at 440 Hz, while the second-chair player is a little off at 442 Hz. What is the beat frequency?