16-4 The Wave Equation

16-4 The Wave Equation#

Prompts

  • What differential equation governs waves on a string? Write it in terms of \(\partial^2 y/\partial x^2\) and \(\partial^2 y/\partial t^2\).

  • How does applying Newton’s second law to a string element lead to the wave equation? What role do the tension forces play?

  • Verify that \(y(x,t) = y_m \sin(kx - \omega t)\) satisfies the wave equation. What condition relates \(k\), \(\omega\), and \(v\)?

  • The wave equation is linear. What does that imply when two waves overlap on the same string?

Lecture Notes#

Overview#

  • The wave equation is the differential equation that governs wave propagation. For a string, it follows from Newton’s second law applied to a string element.

  • Key idea: The curvature of the string (second derivative in \(x\)) produces a net force; that force accelerates the element (second derivative in \(t\)). The wave speed \(v\) ties the two together.

  • The equation is linear—solutions can be added (superposition), which underlies interference (section 16-5).

Deriving the wave equation#

Consider a short string element of length \(dx\) and mass \(dm = \mu\,dx\) as a wave passes. The tension \(\tau\) pulls at both ends.

  • Forces: At each end, the tension force has a vertical component \(F_y = \tau S\), where \(S = dy/dx\) is the slope. The slopes differ slightly at the two ends because of the curvature.

  • Net force: \(F_{2y} - F_{1y} = \tau(S_2 - S_1) = \tau\,\frac{dS}{dx}dx = \tau\,\frac{\partial^2 y}{\partial x^2}dx\).

  • Newton’s second law: \(\mu\,dx\,\frac{\partial^2 y}{\partial t^2} = \tau\,\frac{\partial^2 y}{\partial x^2}dx\).

Rearranging and using \(v = \sqrt{\tau/\mu}\):

(98)#\[ \frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2}\,\frac{\partial^2 y}{\partial t^2} \]

The wave equation#

  • Physical meaning: The spatial curvature (\(\partial^2 y/\partial x^2\)) drives the temporal acceleration (\(\partial^2 y/\partial t^2\)). The factor \(1/v^2\) sets the wave speed.

  • Any function of the form \(y(x,t) = h(kx \pm \omega t)\) with \(v = \omega/k\) satisfies this equation—it describes traveling waves of arbitrary shape.

Checking a sinusoidal wave

For \(y = y_m \sin(kx - \omega t)\): \(\partial^2 y/\partial x^2 = -k^2 y\) and \(\partial^2 y/\partial t^2 = -\omega^2 y\). The wave equation gives \(-k^2 = -\omega^2/v^2\), so \(v = \omega/k\)—consistent with section 16-1.

Superposition#

The wave equation is linear (no \(y^2\) or other nonlinear terms). Therefore, if \(y_1\) and \(y_2\) are solutions, so is \(y_1 + y_2\).

Principle of superposition: When two waves overlap, the net displacement is the sum of the individual displacements:

(99)#\[ y(x,t) = y_1(x,t) + y_2(x,t) \]

  • Overlapping waves add; they do not alter each other’s propagation.

  • This principle underlies interference (section 16-5).

Summary#

  • Wave equation: \(\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2}\,\frac{\partial^2 y}{\partial t^2}\)—derived from Newton’s second law for a string element.

  • Traveling waves \(y = h(kx \pm \omega t)\) with \(v = \omega/k\) are solutions.

  • Superposition: Linear equation \(\Rightarrow\) waves add when they overlap.