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The Linear Wave Equation

All wave functions y(x, t) represent solutions of an equation called the linear wave equation. This equation gives a complete description of the wave motion, and from it one can derive an expression for the wave speed. Furthermore, the linear wave equation is basic to many forms of wave motion. In this section, we derive this equation as applied to waves on strings. Suppose a traveling wave is propagating along a string that is under a tension T. Let us consider one small string segment of length Δx (Fig. 16.22). The ends of the segment make small angles θA and θB with the x axis. The net force acting on the segment in the vertical direction is
linear wave equation Because the angles are small, we can use the small-angle approximation sin θ ≈ tan θ to express the net force as
linear wave equation
linear wave equation However, the tangents of the angles at A and B are defined as the slopes of the string segment at these points. Because the slope of a curve is given by ∂y/∂x we have
linear wave equation We now apply Newton’s second law to the segment, with the mass of the segment given by m= μ Δx :
linear wave equation Combining Equation 16.22 with Equation 16.23, we obtain
linear wave equation The right side of this equation can be expressed in a different form if we note that the partial derivative of any function is defined as
linear wave equation If we associate f(x + Δx) with (∂y/∂x)B and f(x) with (∂y/∂x)A we see that, in the limit Δx → 0, Equation 16.24 becomes
linear wave equation This is the linear wave equation as it applies to waves on a string. We now show that the sinusoidal wave function (Eq. 16.11) represents a solution of the linear wave equation. If we take the sinusoidal wave function to be of the form y(x,t) = Asin( kx–ωt), then the appropriate derivatives are
linear wave equation Substituting these expressions into Equation 16.25, we obtain
linear wave equation This equation must be true for all values of the variables x and t in order for the sinusoidal wave function to be a solution of the wave equation. Both sides of the equation depend on x and t through the same function sin( kx – ωt ). Because this function divides out, we do indeed have an identity, provided that
linear wave equation Using the relationship ν = ω / k (Eq. 16.13) in this expression, we see that
linear wave equation Which is Equation 16.4. This derivation represents another proof of the expression for the wave speed on a taut string. The linear wave equation (Eq. 16.25) is often written in the form
linear wave equation in general This expression applies in general to various types of traveling waves. For waves on strings, y represents the vertical displacement of the string. For sound waves, y corresponds to displacement of air molecules from equilibrium or variations in either the pressure or the density of the gas through which the sound waves are propagating. In the case of electromagnetic waves, y corresponds to electric or magnetic field components.
We have shown that the sinusoidal wave function (Eq. 16.11) is one solution of the linear wave equation (Eq. 16.26). Although we do not prove it here, the linear wave equation is satisfied by any wave function having the form y = f(x ± νt ). Furthermore, we have seen that the linear wave equation is a direct consequence of Newton’s second law applied to any segment of the string.

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