A Transverse Wave on a String – Part One – The Wave Equation

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 \(\eta\).

Figure 1 - Initial Displacement of a taut string

Figure 1 – Initial Displacement of a taut string

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

Figure 2 – Relationship for each angle relating the change in \(\eta\) and the change in \(x\)

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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