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Unformatted text preview: MECHANICAL WAVES (TRAVELLING) We begin our discussion of the wave phenomenon by considering waves in matter. The
simplest deﬁnition of a wave is to call it a traveling disturbance (or equivalently, deviation from
equilibrium). For instance, if you drop a stone on the surface of an undisturbed body of water
you can watch the “disturbance” traveling radially out of the “point” of contact. Formally, we can “construct” a wave in several steps. For simplicity, we take a wave
traveling along x—axis. Step 1. We need a disturbance D.
Step 2. D must be a function of x. Step 3. D must also be a function of t.
Step 4. If x and t appear in the function in the combinations (x Trvt) the disturbance D cannot be stationary. It must travel along x with speed v. Further, (x—vt) implies v = vﬂtravel in+ ivex — direction]
—) (x+vt) implies v = —v£[travel in— we): — direction]
._) EXERCIZE: Put D = A(x — If)2 and show that “parabola” travels. Periodic Waves The simplest wave is when (x-vt) appears in a sin or cos function. D = sin (x—vt) But this
equation is not justiﬁed. First, since D is a disturbance it must have dimensions so we need D: A Sin(x— vt) Where A has the dimensions of D. Next, argument of Sin cannot have dimensions, so we need (x— vt)
/l D=A Sin v Where 1 is alength. Since A has dimension of (l/Time), put is: % 275x 2m Next, introduce a phase angle Q and we get D = A Sin(7— 7+ Q) as the most general periodic wave. Note that 27: has been put in, as we know repeat angle for Sin. If you put
Q = 7? you recover the Equation in some books. . 2m 27m)
D— ASm[ T — A As shown in class xt = Repeat Distance= wavelength
T = perlod, E = f (frequency) And v = ftf 27$ Next, deﬁne k = 7 (wave vector) 0) = 27tf (angular frequency)
a) = vk And we can write D = A Sin (kx — wt + Q) for any periodic wave traveling a long +z've x—axis A x H a)
with velocity v I
4, Similarly, D = A Sin (be + wt + Q) is any periodic wave along —ive x-axis with ...
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- Fall '11