Derivation of Shear Wave Speed

Deriving the Wave Equation from Shear Modulus

The wave equation for shear waves (transverse waves in a solid) can be derived using the shear modulus ( $G$ ), which relates shear stress to shear strain, and Newton’s second law. Below is a step-by-step derivation, focusing on a shear wave propagating in a continuous medium.

Step 1: Shear Stress and Strain Relationship

The shear modulus ( $G$ ) defines the relationship between shear stress ( $τ$ ) and shear strain ( $γ$ ):

τ = G \cdot γ

For a small deformation in a solid, if a shear wave displaces particles in the $y$ -direction while propagating along the $x$ -direction, the shear strain is the spatial derivative of the displacement $u (x, t)$ :

γ = \frac{\partial u}{\partial x}

Thus, the shear stress is:

τ = G \frac{\partial u}{\partial x}

Step 2: Applying Newton’s Second Law

Consider a small element of the medium with mass $d m = ρ A d x$ , where $ρ$ is the density, $A$ is the cross-sectional area, and $d x$ is the length along the $x$ -direction.

The net force on this element arises from the difference in shear stress across its faces:

At position $x$ , the shear stress is $τ = G \frac{\partial u}{\partial x}$ .
At position $x + d x$ , the shear stress is $τ + \frac{\partial τ}{\partial x} d x = G \frac{\partial u}{\partial x} + G \frac{\partial^{2} u}{\partial x^{2}} d x$ .

The net force $F$ in the $y$ -direction is:

F = (G \frac{\partial u}{\partial x} + G \frac{\partial^{2} u}{\partial x^{2}} d x) A - (G \frac{\partial u}{\partial x}) A

F = G \frac{\partial^{2} u}{\partial x^{2}} A d x

Using Newton’s second law ( $F = m a$ ), where the acceleration is $a = \frac{\partial^{2} u}{\partial t^{2}}$ :

ρ A d x \frac{\partial^{2} u}{\partial t^{2}} = G \frac{\partial^{2} u}{\partial x^{2}} A d x

Step 3: Simplifying to the Wave Equation

Cancel $A d x$ (assuming $A$ is constant and $d x \neq 0$ ):

ρ \frac{\partial^{2} u}{\partial t^{2}} = G \frac{\partial^{2} u}{\partial x^{2}}

Rearrange:

\frac{\partial^{2} u}{\partial t^{2}} = \frac{G}{ρ} \frac{\partial^{2} u}{\partial x^{2}}

This is the one-dimensional wave equation, where the wave speed $v$ is:

v = \sqrt{\frac{G}{ρ}}

Physical Insight

The shear modulus ( $G$ ) quantifies the medium’s resistance to shear deformation.
The density ( $ρ$ ) represents the inertia of the medium.
The wave speed $v$ depends on the ratio $G / ρ$ , consistent with physical intuition that stiffer materials (higher $G$ ) propagate waves faster, while denser materials (higher $ρ$ ) slow them down.

Unfortunately, the provided reference materials do not explicitly contain this derivation. However, the general approach aligns with standard physics principles found in texts like "Physics for Scientists and Engineers" by Giancoli, which discusses Newton’s laws and wave propagation^[1].

Newton’s second law is used to derive equations of motion. Reference: "Physics for Scientists and Engineers," p.88-91. ↩︎