FACULTY OF SCIENCE, AUTUMN 2007
PHYS 124 LEC A1 : Particles and Waves (Instructor: Marc de Montigny)
Walker, Physics, Chapter 14: Waves and Sound
Some sections will be dropped, due to lack of time.
- Section 14-1: Types of Waves
- P. 431 : A wave is a disturbance that propagates energy from one place to another. This disturbance is represented by a Wave Function, which describes, for instance, the transverse displacement of a point along a string (Section 14-2), the longitudinal displacement of the rings of a Slinky (P. 432), the vertical displacement of the points of the surface of water, the air pressure of sound waves (Section 14-4), the amplitude of electric and magnetic fields of light waves (Chapters 25, 28), the probability amplitude of a particle in Quantum Physics (Chapter 30), etc.
- Be careful to distinguish between (1) the motion of the wave itself, and (2) the motion of the individual particles involved in the disturbance.
- P. 431: Transverse Waves (individual particles have a transverse motion w/r to the wave propagation) Figure 14-1, Figure 14-2, Figure 14-4, Figure 14-5. See also the Wave on a String simulation.
- P. 433, Wavelength λ = distance over which a wave repeats (or, spatial length of a periodic cycle). Analogous to the Period T, which is the temporal length of a cycle.
- P. 433, Eq. 14-1 : v = distance/time = λ/T = λ f (Speed of a Wave).
- This can be understood by thinking of a train consisting of wagons with length λ. If the wagons pass by at a rate of T seconds per wagon, then we obtain Eq. 14-1.
- P. 433 Figure 14-6
- P. 433, Exercise 14-1
- Section 14-2: Waves on a String
- P. 434, Eq. 14-2 : v = (F/μ)1/2 (Speed of a Wave on a String)
- Mass per Length μ = m/L
- Reflections: P. 436, Figure 14-7 (Fixed End), Figure 14-8 (Free End).
- Section 14-3: Harmonic Wave Functions
- OMITTED : NOT IN SYLLABUS
- Section 14-4: Sound Waves
- P. 438 : Sound Waves are longitudinal, like the waves on a Slinky. Figure 14-11 shows (a) the rarefactions and compressions of a typical sound wave, (b) fluctuations of the density of the air versus x, (c) fluctuations in the pressure of air as a function of x.
- P. 438, Table 14-1 shows the speed of sound in various materials.
- Human ear perceive sound waves with frequency 20 < f < 20,000 Hz. Sounds with f > 20,000 Hz are ultrasonic, and sounds with f < 20 Hz are infrasonic.
- Section 14-5: Sound Intensity
- P. 442, Eq. 14-5 : I = P/A (Sound Intensity, P = E/t is the power (in Watt), and A (in m²) is the area through which the sound wave go.)
- P. 441, Figure 14-12
- P. 442, Table 14-2 contains various sound intensities.
- P. 443, Eq. 14-7 : I = P/4πr² (Units : W/m²)
- P. 471, Problem 32
- P. 471, Problem 33
- P. 444, read Section "Human Perception of Sound". Between the Threshold of Hearing and the Threshold of Pain, I covers 12 orders of magnitude; this suggests the use of a logarithmic scale of intensity : the decibels (omitted in the course).
- Section 14-6: The Doppler Effect
- OMITTED due to lack of time
- Section 14-7: Superposition and Interference
- P. 452, Principle of Linear Superposition : y = y1 + y2 (not valid in general, but only when the equations which describe the waves are linear).
- P. 452, Figure 14-19 illustrates the superposition.
- P. 452, Figure 14-20 illustrates constructive interference and destructive interference.
- Bottom of P. 453: (Δd = difference of path lengths)
- Constructive interference : Δd = 0, λ, 2λ, 3λ,... = mλ (m integer)
- Destructive interference : Δd = λ /2, 3λ /2, 5λ /2,... = (m+1/2)λ (m integer)
- P. 472, Problem 53
- Section 14-8: Standing Waves
- 14-8-1 : Standing Waves on a String, PP. 456-458
- P. 456, Figure 14-23 shows the fundamental mode of a standing wave on a string with both ends fixed.
- P. 456, Figure 14-24 shows higher frequencies. (A : antinode, N : node). One observes that
- The FIRST (or fundamental) harmonic has λ1 = 2L, and f1 = v / λ1 = v / 2L (P. 457, Eq. 14-12)
- The second harmonic has λ2 = 2L/2 = L, so that f2 = v / λ2 = v / L = 2 f1
- The third harmonic has λ3 = 2L/3, so that f3 = v / λ3 = 3v / 2L = 3 f1
- P. 457, Eq. 14-13 : fn = n f1 = nv /2L (n = 1, 2, 3,...); λn = λ1 /n = 2L/n
- P. 459, Figure 14-25 explains why the distance between a guitar's frets is not uniform : to go up one octave from the fundamental, the effective length of a string must be halved. To increase the second octave, once must halve the length once again, etc. The frets are therefore more closely spaced near the base of the neck.
- 14-8-2 : Vibrating Columns of Air, PP. 459-462
- Standing Longitudinal Waves
- 14-8-2(a) : Pipe Open at Both Ends (similar to the waves on a string)
- P. 461, Figure 14-29 shows the standing waves in a pipe that is open at both ends
- P. 462, Eq. 14-15 : f1 = v / 2L ; fn = n f1 = nv /2L (n = 1, 2, 3,...); λn = λ1 /n = 2L/n
- 14-8-2(b) : Pipe Open at One End
- P. 459, Figure 14-26 : exciting a standing wave. (A : antinode, N : node)
- P. 460, Figure 14-27 shows the first harmonics. One observes that
- The FIRST (or fundamental) harmonic has λ1 = 4L, and f1 = v / λ1 = v / 4L (bottom of P. 459)
- The THIRD (n = 3) harmonic has λ3 = 4L/3, so that f3 = v / λ3 = 3v / 4L = 3 f1
- The FIFTH (n = 5) harmonic has λ5 = 4L/5, so that f5 = v / λ5 = 5v / 4L = 5 f1
- P. 460, Eq. 14-14 : fn = n f1 = nv /4L (n = 1, 3, 5,...); λn = λ1 /n = 4L/n
- P. 472, Problem 58
- P. 472, Problem 63
- P. 472, Problem 64
- P. 472, Problem 65
- In real life, sounds are superposition of different modes, as shown in the following simulation: Fourier: Making Waves. The tone, or sound quality, is determined by the relative amplitudes of frequencies in the spectrum.
- Frequency of Musical Tones
- Section 14-9: Beats
- OMITTED due to lack of time