1 Part 5: Waves 5.1: Harmonic Waves Wave a disturbance in a medium that propagates Transverse wave - the disturbance is perpendicular to the propagation direction (e.g., wave on a string) Longitudinal wave the disturbance is parallel to the propagation direction (e.g., sound wave) 1-D Wave Equation & Harmonic Waves For a wave traveling along the x-axis, the disturbance y at some position x at some time t must satisfy 2 y x 2 = 1 2 y v 2 t 2 One set of solutions to the wave equation are harmonic waves that have a sine or cosine shape. For a harmonic wave, v is the speed of the wave (phase velocity). The phase velocity depends on properties of the medium through which the wave travels. Example: wave on a string v = F μ F = tension in string [N] = linear mass density (mass per length) [kg/m] A harmonic wave function can be written using either a sine or cosine function y(x, t) = A cos(kx ± ωt + φ) or y(x, t) = A sin(kx ± ωt + φ) A= amplitude of wave [mks units of the disturbance y] k = wave number [rad/m] = angular frequency [rad/s] = phase angle [rad]
2 A plot of the disturbance versus position at some time looks like: y 0 A x = wavelength = distance between consecutive crests [m] k = 2 / [rad/m] A plot of the disturbance versus time at one position along the wave looks like: y T 0 A t T = period = time for one oscillation or cycle [s] f = 1 / T = frequency = number of cycles per time [cycles/s = Hz] = 2 / T = 2 f [rad/s] The sign between kx and t determines the direction the wave travels along the x-axis. + wave travels to left (in the direction of decreasing x) - wave travels to right (in the direction of increasing x) The phase angle shifts the cosine or sine function left or right. This can be used to match some initial condition for the wave function. Phase velocity v = f = ω k Power delivered by wave is proportional to the speed and the squares of the amplitude and frequency. P A 2 f 2 v
3 5.2: Sound Waves A sound wave is a vibration of the layers of a medium that travels as a longitudinal wave. Speed of sound in calm air: v 343 m/s 767 mph The frequency of the wave determines the pitch of the sound wave. Lower frequency gives lower pitch. 767 Typical human audible range: 20 20,000 Hz (20 Hz 20kHz) A. Power & Intensity A sound wave delivers energy at a certain rate (the power P) and this energy is spread out over some area A. ***NOTE: In this section on sound, A is this area, not the amplitude of the sound wave. The intensity of the wave is the power per area: I = P A mks units [Watts per square meter = W/m 2 ] The intensity of the sound determines the loudness of the sound. Threshold of human hearing: 10-12 W/m 2 Threshold of pain: 1 W/m 2 This intensity range of 12 orders of magnitude can be decreased by converting the intensity to the sound level and using the decibel scale. Converting from intensity to sound level: β = 10 log I 10 12 I = intensity in [W/m 2 ] = sound level [decibel = db] This shrinks the range from the threshold of hearing to the threshold of pain to 0 to 120 db. Note that the db is a unitless unit. The sound level is a number. We tack on the db to inform people that this number refers to a sound level.
4 Converting from sound level to intensity: ( 12+β 10) I = 10 = sound level [decibel = db] I = intensity in [W/m 2 ] Some sources of sound can be approximated as point sources. A point source sends out a wave uniformly in all directions. The energy is spread out uniformly across spherical surfaces. The surface area of a sphere of radius r is 4 r 2. Thus, the intensity for a point source is I = P 4πr 2 For two different distances from a point source I 1 = r 2 2 I 2 2 r 1 x r Example: Ten meters from a jet engine, the sound level is 150 db. How far from the engine is the level reduced to 50 db if you assume that the engine is a point source of sound? Does this answer seem realistic? Why is this answer so large? Ans. 1000 km = 620 miles
5 B. Doppler Effect If there is relative motion between a sound source and an observer, then the observed frequency is shifted from the source frequency. If the distance between source and observer is decreasing, the frequency shifts up. If the distance between source and observer is increasing, the frequency shifts down. Doppler Shift Equation f = f ( v ± v o v v s ) f = observed frequency f = source frequency v = speed of sound (in calm air v = 343 m/s = 767 mph) v o = speed of observer v s = speed of source Numerator sign choice: If the observer travels towards the source, the top sign is used (+). If the observer travels away from the source, use the bottom sign (-). Denominator sign choice: If the source travels towards the observer, the top sign is used (-). If the source travels away from the observer, use the bottom sign (+).
6 5.3: Wave Interference When two waves overlap in some region, the resulting wave is the superposition of the waves. If this superposition pattern doesn t change for a reasonable time, then the superposition forms an interference pattern. This pattern has points with varying amounts of intensity. An intensity maximum occurs when the waves are exactly in phase. (crest on crest and trough on trough) An intensity minimum occurs when the waves are out of phase by /2. (crest on trough) Interference Example: Standing Waves on a String The string is fixed at both ends. Waves reflect off the fixed ends and interfere with each other to form standing wave patterns at certain frequencies and, thus, at certain wavelengths. Standing wave wavelengths: n = 2L n Standing wave frequencies: L = length between fixed ends n = 1, 2, 3 f n = v λ = n v 2L f 1 = v / 2L f 2 = 2f 1 f 3 =3f 1 fundamental frequency first harmonic second harmonic node antinode node antinode 1 = 2L 2 = L 3 = 2L/3
7 5.4 Electromagnetic Waves An electromagnetic wave (light wave) consists of oscillating electric and magnetic fields. The directions of the electric and magnetic fields are perpendicular. The wave travels in the third perpendicular direction. It is a transverse wave. The fields store energy and transport the energy in the wave. For example, the picture below shows the wave traveling in the +x direction. The electric field oscillates in the y direction and the magnetic field oscillates in the z direction. The individual electric and magnetic waves are in phase. The fields peak at the same position at the same time. Using Faraday s Law and the Ampere-Maxell Law for the fields shown above traveling in free space (vacuum), one can get the following wave equations for the fields. 2 E(x, t) x 2 2 B(x, t) x 2 = μ o ε o 2 E(x, t) t 2 = μ o ε o 2 B(x, t) t 2 The harmonic solutions to these equations are of the form E(x, t) = E M cos(kx ωt + φ) B(x, t) = B M cos(kx ωt + φ) Amplitudes: E M mks {V/m] B M mks [T]
8 Speed of Light in Free Space: c = 3x10 8 m/s = 186,000 mph v = c = 1 μ o ε o Relationship between speed and fields: Power and Intensity: v = E M E(x, t) = B M B(x, t) The power is the rate that energy is carried in the wave. P E M B M f 2 v E M 2 f 2 The power is spread out over an area A. The intensity is power per area. It determines the brightness of the wave. I = P A Example: The Sun emits power in all directions. At Earth, the received intensity is approximately 1370 W/m 2. This is known as the solar constant. Electromagnetic Spectrum For a harmonic light wave in free space: c = o f o = free-space wavelength The frequency range of light waves is broken into different bands to for the electromagnetic spectrum. See more detailed information about the spectrum on the course web page.