2 Types of Mechanical Waves A mechanical wave is a wave that requires a mediumTransverse waves – the displacement of the medium is perpendicular to the direction the wave travelsLongitudinal waves – the motion of the medium is back and forth in the same direction as the waveWave speed ≠ particle speedWaves are disturbances; transport energy (but not matter) from one region to another
3 Periodic Waves Periodic/Sinusoidal wave: Wave has constant velocity, but every particle undergoes simple harmonic motionWavelength – distance from point on one wave shape to identical pt. on the nextFrequency - number of cycles per unit time:Velocity:rarefactioncompressionλ
4 Mathematical Description wave functionWave equation:Model for any wave (periodic or non-periodic, electromagnetic, sound, on string, in water…)
5 Speed of Transverse Waves and Wave Energy (speed of wave on string)F is tension and μ is mass per unit lengthWaves transport energy from one place to another(avg. power of wave on string)Intensity – the average rate at which energy is transported by the wave, per unit of surface area(inverse-square law for intensity)
6 Reflection Boundary conditions – whether the end is fixed (right) or free (left)
7 Interference and Superposition Interference – two or more waves passing through the same regionPrinciple of Superposition – when two waves overlap, the displacement at any pt is the sum of the individual waves’ displacements
9 Standing WavesPattern resulting from the combination of the reflected wave and the original waveNodes – pts where amplitude never changesAntinodes – pts where string fluctuates up to the amplitude and then back down with each passing wave bodyDoes not appear to move, so is called a standing wave (as opposed to a traveling wave – last slide)
10 ConstructiveinterferencedestructiveStanding WavesOperate based on the principle of superposition (see right)Nodes occur at x=0, λ/2, λ, 3λ/2,…Where do antinodes occur?
11 Normal Modes [n=1,2,3,…] (standing wave, fixed at both ends) L is length of string, n is # of antinodes[n=1,2,3,…]f1 is the fundamental frequencyThese frequencies are harmonics within the harmonic seriesf2 is the 2nd harmonic and 1st overtone; f3 is the third harmonic and the second overtone
12 Free ResponseIf a string forms a standing wave with 3 antinodes and a wavelength of 4 cm, how long is the string?A standing wave travels at 5 m/s on a 1 m long string. What is the frequency of the 2nd overtone?
13 Normal Modes (string fixed at both ends) Normal mode – motion in which all particles in a system move sinusoidally with the same frequencysee previous diagramnot the case for musical instrumentsWhat are timbre and harmonic content?
15 Sound Waves Sound is a longitudinal wave 20 – 20,000 Hz is the audible rangeDisplacement amplitude – the max. displacement of a particle from its equilibrium positionUseful to describe sound waves in terms of pressure differencesPressure amplitude = max pressure fluctuation =
16 Speed of Sound In a fluid: In a solid rod: In an ideal gas: At 20°C, speed of sound in air is 344 m/sDoes sound travel fastest in a gas, a liquid, or a solid? Slowest?
17 Sound IntensityA low-frequency sound need a large amplitude to have the same intensity as a high-frequency sound (b/c )Decibel scalelogarithmic scale+10 dB = x 10 intensity; +20 dB = x 100 intens.
18 Free ResponseA concert is recorded as having a decibel level of 170 dB. Since I0 is 10^-12 W/m², what is the intensity of the sound?If one person speaks at 8x10^-4 W/m² and a second person yells at 4x10^-3 W/m², how many decibels louder is the second?
19 Standing Sound WavesIn same way that transverse standing waves form, longitudinal standing waves can formPressure nodes vs. Displacement nodesPressure node is a point in a standing sound wave at which pressure and density do not varyPressure antinode is a point at which pressure and density vary the greatestan
20 Open Resonators Open pipe – open at both ends Displacement nodes are at each endEx – organ, flute, recorder, ocarina?
21 Stopped Resonators Stopped pipe – one open end, one closed Antinode at the open end and node at closed endEx – Oboe, clarinet
22 Resonance Similar to concept of driven oscillation If the frequency of a speaker, voice, etc. matches one of the normal-mode frequencies of resonator (i.e. pipe), then the resonator vibrates with maximum amplitude.Aretha Franklin example
23 InterferenceConstructive interference – two or more waves meet so that the resulting wave is larger than either of the originals (i.e. crest to crest, trough to trough)Waves are in phaseDestructive Interference – two or more waves seem to cancel each other out (i.e. crest meets trough)Waves are out of phase
24 InterferenceConstructive/ in phase: waves differ by an integer multiple of λ (λ, 2λ, 3λ, …)Destructive/ out of phase: waves differ by an integer multiple of λ/2 (λ/2, 3λ/2, 5λ/2,…)
25 Free ResponseTwo loudspeakers are positioned as below. The both produce a frequency of 784 Hz. The speed of sound in air is 344 m/s. a) At what distances from B will there be destructive interference? b) What distances will produce constructive interference? c) If the frequency is made low enough, there will be no positions along the line BC at which destructive interference occurs. How low must this frequency be?
26 BeatsWhen two sounds destructively interfere slightly out of phase, the resulting superposition (wave) has a different frequency, the beat frequency
27 Doppler EffectSimply – when a source of sound and a listener are in relative motion, the perceived frequency differs from the actualThink of a boat moving towards and away from shoreApplies to light as well – red shifting
28 Free ResponseA car alarm is emitting sound waves of frequency 520 Hz. You are on a motorcycle, traveling directly away from the car. How fast must you be traveling if you detect a frequency of 490 Hz?