Summary and Concept Map

A wave is a disturbance which carries energy and momentum from one point in space to another point in space without the transfer of the medium; mechanical waves require a medium for their propagation, while non-mechanical waves do not require a medium. For a transverse wave, particle vibration is perpendicular to the direction of propagation of the wave; for a longitudinal wave, particle vibration is parallel to the direction of propagation. Elasticity and inertia are necessary properties of the medium for wave propagation -- ripples on still water are transverse, while the wave formed by a vibrating tuning fork is longitudinal. The distance between two consecutive crests or troughs is the wavelength $\lambda$; the number of waves crossing a point per second is the frequency $f$; the time taken by one wave to cross a point is the time period $T$; and the wave velocity is $v=f\lambda$, with frequency being source-dependent and wave velocity being medium-dependent. The velocity of a transverse wave on a stretched string is $v=\sqrt{T/\mu}$, independent of the shape of the wave, and the velocity of a longitudinal wave in an elastic medium is $v=\sqrt{E/\rho}$. The minimum distance from a sound-reflecting wall needed to hear a distinct echo at 20 degC is 17.2 metres. The one-dimensional wave equation is $\partial^2y/\partial x^2=(1/v^2)\,\partial^2y/\partial t^2$, and the wave number is $k=2\pi/\lambda$. During interference, the resultant intensity is $I=I_1+I_2+2\sqrt{I_1I_2}\cos\phi$ (since intensity is the square of amplitude, $I=A^2$); constructive interference gives $I_{max}=(\sqrt{I_1}+\sqrt{I_2})^2=(A_1+A_2)^2$, and destructive interference gives $I_{min}=(\sqrt{I_1}-\sqrt{I_2})^2=(A_1-A_2)^2$. Superposing two or more waves of slightly different frequency produces beats, with beat frequency equal to the difference of the two source frequencies. If natural frequencies are written as integer multiples of the fundamental, they are said to be in harmonics: the first harmonic is the fundamental, the second harmonic is twice the fundamental, and so on. Loudness is the degree of sensation of sound produced in the ear, and intensity is the sound power transmitted per unit area normal to the direction of propagation; the sound intensity level is $\Delta L=10\log_{10}(I_1/I_0)$ decibel. A closed organ pipe has only odd harmonics, $f_n=(2n+1-2)f_1$ in the ratio $1:3:5:7:\ldots$, while an open organ pipe has all harmonics, $f_n=nf_1$, in the ratio $1:2:3:4:\ldots$. Whenever there is relative motion between a sound source and a listener, the observed fre …