Chapter2

Chapter2 - Waves and the One-Dimensional Wave Equation...

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Waves and the One-Dimensional Wave Equation Earlier we talked about the waves on a pond. Before we start looking specifically at sound waves, let’s review some general information about waves. Types There are two general classifications of waves, longitudinal and transverse: Transverse Wave – A traveling wave in which the particles of the disturbed medium move perpendicularly to the wave velocity. An example is the wave pulse on a stretched rope that occurs when the rope is moved quickly up and down. Longitudinal Wave – A traveling wave in which the particles of the medium undergo displacement parallel to the direction of the wave motion. Sound waves are longitudinal waves. One thing to note is that some waves exhibit characteristics of both types of waves. The waves on our pond are a combination of both types. Transverse: Longitudinal: 2-1

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Characteristics Just like the periodic motion of the simple harmonic oscillator, waves have certain characteristics. The ones we will concentrate on are the frequency, period, wave speed and the wavelength. Recall from SP211, a picture of a transverse wave in a medium at some time, maybe t=0 sec. Traveling Wave at t = 0 -1.5 -1 -0.5 0 0.5 1 1.5 0246 x (m) s (m) λ s 0 We wrote an equation to describe this picture: () 0 2 sx ss i n x π ⎛⎞ = ⎜⎟ λ ⎝⎠ where: s = particle displacement – Distance that the fluid particle is moved from its equilibrium position at any time, t. s o = maximum particle displacement or amplitude λ = distance over which the wave begins to repeat k = 2 π λ = a conversion factor that relates the change in phase (angle) to a spatial displacement. We call k the wavenumber. When we let this wave begin to move to the right with a speed, c, the position is shifted in the governing equation from x to x-ct. o 2 , t i n x c t π =− λ 2-2
Below is a picture of the same traveling wave shown at some later time, t. Traveling Wave at Some Later Time, t -1.5 -1 -0.5 0 0.5 1 1.5 02468 x (m) s (m) ct Now, if instead of taking a snap shot of the wave in the medium at two different times, what if we had set a sensor somewhere in space – maybe at x = 0 m, and recorded the wave’s displacement over time. The equation governing the wave would become: () [] oo o 22 c 2 s0 , t ss i n 0 c t i n t i n t i n t T ππ π ⎡⎤ = =− ω ⎢⎥ λλ ⎣⎦ o where T = period – Time to complete one cycle. c = T λ = wave velocity – Distance that wave energy travels per unit time. ω = 2 T π = a conversion factor that relates the change in phase (angle) to a temporal displacement. We call ω the angular frequency. f = 1 T = frequency, is the inverse of the period. It is the number of cycles per unit time that pass the origin. Note that we have employed a similar strategy regarding the group of constants in front of the time variable that we used when discussing the wavenumber, k. Since the wave repeats every 2 π change in phase and that corresponds to a time period, T, angular frequency, ω =2 π /T, is nothing more than a conversion factor from time to phase angle.

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Chapter2 - Waves and the One-Dimensional Wave Equation...

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