1. Physics 141Mechanics Lecture 22 Waves Yongli Gao The propagation of oscillations, such as ripples on a pond, sound, radio, TV, is a physical phenomenon called wave. There're three main types of waves: mechanical, electromagnetic, and matter waves. We'll take mechanical waves to describe most of the characteristics of waves. Waves transfer energy, not matter. If displacement of particles is perpendicular to the direction of wave propagation, it is a transverse wave. If displacement of particles is parallel to the direction of wave propagation, it is a longitudinal wave.

2. Period and Frequency The period T of a wave is the time taken for the oscillating motion of any point to repeat itself. T is in second. The frequency f of a wave is the number of complete oscillations per unit time. Clearly The unit for frequency is hertz. 1 hertz = 1/s.

3. Sinusoidal Waves The simplest form of waves are sinusoidal waves. where A is the amplitude of the wave,  is the angular frequency, kx-  t the phase. k is called the wave number. Its unit is 1/m.  (x,t) describes the displacement of a particle in mean position x at time t. Suppose we look at a position x 0, and its motion is just SHO, with frequency, or period

4. If we take a snapshot of the wave at time t 0, we see the periodicity in space. The distance between two successive peaks is called the wavelength. Wavelength is the spatial analog of period. Note that the wave only depends on the phase kx-  t. A particle in the wave only oscillates about its average position in a SHO motion. On the other hand, if you follow a feature of the wave, say, a peak, it will appear to move at a constant velocity, which is called phase velocity.

5. Superposition of Waves When two waves overlap, the actual displacement is the sum of two displacements, one for each wave as if the other one were not present. Principle of superposition: suppose two waves are traveling along the same stretched string, In fact, any curve can be expressed uniquely as the superposition of sinusoidal curves. Finding out such a combination is called Fourier analysis.

6. Interference of Waves If two waves are of the same frequency, they may enhance or diminish each other, depending on the relative phase of the two waves, or, they interfer. Interference can occur in any dimension. Suppose two waves are propagating in the same direction, If A=B

7. Standing Waves If two waves are of the same frequency and amplitude, but propagating in opposite directions, This is a standing wave. At positions such that the amplitude is always zero. These points are called the nodes of the standing wave. In terms of the wavelength,

8. In terms of the wavelength, The distance between two adjacent notes is /2. For a string of length L and with both ends fixed, the ends must be two of the nodes. Possible standing waves are of wavelengths 1 =2L (2 nodes at the ends) 2 =L (3 nodes, 1 at the center, 2 at the ends) 3 =2L/3 (4 nodes, 2 at the center, 2 at the ends) … These are called the normal modes of the string, or eigen modes.