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**Unformatted text preview: **ing waves are possible only for certain wavelengths, then only a few specific oscillation
frequencies are allowed. Because the wave speed for a sinusoidal wave is 6 (23)
The resonance frequency for these wavelengths is 0, 1, 2, 3, … 2 (24) Equation (24) tells us that the resonant frequencies are integer multiples of the lowest resonant frequency, (25) 2 Which corresponds to n=1. The oscillation mode with that lowest frequency is called the fundamental
mode or the first harmonic. The second harmonic is the oscillation mode with n=2, the third harmonic is
that with n=3, and so on. The collection of all possible oscillation modes is called the harmonic series,
and n is called the harmonic number of the nth harmonic. The phenomenon of resonance is common to
all oscillating systems and can occur in two and three dimensions.
2.8 Wave Speed on a Stretched String
The speed of a wave is related to the wave’s wavelength and frequency by equation (23), but it is
set by the properties of the medium. If a wave is to travel through a medium such as water, air, steel, or a
stretched string, it must cause the particles of that medium to oscillate as it passes. For that to happen, the
medium must possess both mass (so that there can be kinetic energy) and elasticity (so that there can be
potential energy). Thus, the mediums mass and elasticity properties determine how fast the wave can
travel in the medium. Conversely, it should be possible to calculate the speed through the medium in
terms of these properties.
Let us consider a single symmetrical pulse as that of
Figure 4, moving from left to right along a string with speed
v. For convenience, we choose a reference frame in which the
pulse remains stationary; that is, we run along with the pulse,
keeping it constantly in view. In this frame, the string appears
to move past us, from right to left in Figure4 with speed v.
Consider a small string element of length Δl within
the pulse, as a element that forms an arc of a circle of radius R
and subtending an angle 2θ at the center of that circle. A force
r
τ with a magnitude equal to the tension of the string pulls Figure 4 A symmetrical pulse, viewed from a
the pulse
tangentially on this element at each end. The horizontal reference frame in which move rightis stationary
and the string appears to
to left with
components of these forces cancel, but the vertical speed v. Picture from Fundamentals of Physics
r
th
components add to form a radial restoring force F . In 7 edition Wiley Publishing
magnitude, 2 sin 2 ∆ (26) where we have used the small angle approximation for sinθ.
The mass of the element is given by ∆ ∆ where μ is the string’s linear density....

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