In Physics of Waves by William C. Elmore and Mark A. Heald the book begins by looking at waves with the derivation of the formula for motion of a Transverse Wave on a string. In exploring wave phenomena, this material is a good starting point for comprehending both the physics and the mathematics of waves.

The problem is to find the mathematical formula to describe the motion on a taught string due to a temporary displacement of the string.

The string may be treated as a continuous media of linear density \(\lambda_0\) with a constant tension \(\tau_0\).

The string is displaced in the vertical direction \(\eta\).

The tension makes an angle \(\alpha_1\) to the unperturbed path of the string at the point of \(\eta_1\) displacement and the tension makes an angle \(\alpha_2\) to the unperturbed path of the string at the point of \(\eta_2\) displacement.

$$\tau_0$$ is constant.

$$\alpha_1 << 1$$ and $$\alpha_2 << 1$$

Looking at the sum of the forces in the y-direction:

$$F_y=\tau_0 \sin \alpha_2 – \tau_0 \sin \alpha_1$$

Due to the small angles

$$\sin \alpha_i \approx \tan \alpha_i$$

Noting the relationship in Figure 2

Note from this relationship:

$$\tan \alpha_i = \frac{\partial \eta_i}{\partial {x}}$$

This can be placed into the equation for the force in the y-direction:

$$F_y=\tau_0 \frac{\partial \eta_2}{\partial {x}}-\tau_0 \frac{\partial \eta_1}{\partial {x}}$$

Looking at the sum of the forces in the x-direction:

$$F_x=\tau_0 \cos \alpha_2 – \tau_0 \cos \alpha_1$$

The small angles mean \(\cos \alpha_2 \approx 1\) and \(\cos\alpha_1 \approx 1\)

Thus

$$F_x=1- 1=0$$

The only net force is in the y-direction. Using Newton’s second law all of these equations can be combined.

$$F=ma$$

$$m=\lambda_0 \Delta x$$

$$a = \frac{\partial^2\eta}{\partial t^2}$$

Setting this equal to the force in the y-direction and taking the limit as \(\Delta x\) approaches zero this give the following:

$$\tau_0\left(\frac{\partial \eta_2}{\partial x}-\frac{\partial \eta_1}{\partial x}\right) = \lambda_0 \Delta x \frac{\partial^2 \eta}{\partial t^2}$$

$$\frac{1}{\Delta x}\tau_0 \left(\frac{\partial \eta_2}{\partial x}-\frac{\partial \eta_1}{\partial x}\right) = \lambda_0 \frac{\partial^2 \eta}{\partial t^2}$$

Note taking the limit gives the definition of the derivative which generates the second partial derivative with respect to x.

$$\lim\limits_{\Delta x \to 0} \frac{1}{\Delta x} \tau_0 \left(\frac{\partial \eta_2}{\partial x}-\frac{\partial \eta_1}{\partial x}\right) = \lambda_0 \frac{\partial^2 \eta}{\partial t^2}$$

$$\tau_0 \frac{\partial^2 \eta}{\partial x^2} = \lambda_0 \frac{\partial^2 \eta}{\partial t^2}$$

This is called the wave equation the next post will explore properties of this equation.

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