Derivation of One Dimensional Wave Equation

One dimensional wave as the name suggests prescribes to own space dimension, i.e., the only independent variable present is time. There are various examples of waves, such as sound waves, ocean waves, or vibrations that are produced by musical instruments as well as electromagnetic radiations producing waves. A wave is studied in classical physics in mechanics, sound, and light. A wave can be described as a disturbance that travels through a medium transferring energy. A single disturbance is called a pulse, and a repetitive disturbance is called a periodic wave. The medium is a series of interconnected particles exhibiting wave-like nature. The particles interact with one another, allowing the disturbance or wave to travel through such mediums.

Types of Waves

Waves can generally be categorized into two different types, namely, travelling and stationary waves.

• Travelling waves, for example, sea waves or electromagnetic radiation, are waves that "move", implying that they have a recurrence and are spread through space and time where time is the only independent variable. Another method of depicting this property of "wave development" is related to energy transmission– a wave moves over a set distance. The most significant sorts of travelling waves in regular existence are electromagnetic waves, sound waves, and maybe water waves. It is hard to break down waves spreading out in three measurements, reflecting off items, so we start with the least fascinating instances of waves, those limited to move along a line. We should begin with a rope, similar to a clothesline. You take one end free, holding the rope, and, keeping it extended, wave your hand up and back once. On the off chance that you do it sufficiently quickly, you'll see a solitary knock travel along the rope.

• As opposed to travelling waves, standing waves, or stationary waves, stay in a consistent situation with peaks and boxes in fixed stretches. One method of creating an assortment of standing waves is by pulling a guitar or violin string. While putting one's finger on a part of the string and then pulling the string with another finger, one has made a standing wave. The examples for this wave include the string wavering in a sine-wave design with no vibration at the closures. There is additionally no vibration at a progression of similarly divided focuses between the closures. These "calm" places are hubs. The spots of greatest wavering are antinodes.

The Wave Equation

The One-dimensional wave equation was first discovered by Jean le Rond d'Alembert in 1746. The mathematical representation of the one-dimensional waves (both standing and travelling) can be expressed by the following equation:

\[\frac{\partial^{2} u(x, t)}{\partial x^{2}} \frac{1  \partial^{2} u(x, t)}{v^{2} \partial t^{2}}\]

Where u is the amplitude, of the wave position x and time t, with v as the velocity of the said wave, this equation is known as the linear partial differential equation in one dimension. This equation tells us how 'u' can change as a function of time and space. 

One-Dimensional Wave Equation Derivation

Let us consider the relationship between the volume ∆v in the direction x and Newton's law which is being applied to it:

\[\triangle F = \frac{\triangle mdv x}{dt}\] (Newton's law)

Where F is the force acting on the element with volume ∆v,

\[= \triangle Fx = - \triangle px \triangle Sx = (\frac{\partial p \triangle x}{\partial x} + \frac{\partial p dt}{\partial x}) \triangle Sx \simeq - \triangle V \frac{\partial p}{\partial p}{\partial x} - \triangle V \frac{\partial p}{\partial p}{\partial x} = M \frac{dvx}{dt}\]

dt is minuscule; therefore it is not considered, and ΔSx is in the x-direction, so, ΔyΔz and from Newton’s law).

\[ = \frac{\rho \triangle V dvx}{dt}\]

From,

\[\frac{dvx}{dt}\] as \[\frac{\partial vx}{dt} \frac{dvx}{dt} = \frac{\partial vx}{\partial dt} + vx \frac{\partial vx}{\partial x} \approx \frac{\partial vx}{\partial x} - \frac{\partial p}{\partial x} = \rho \frac{\partial vx}{\partial t}\] (This is the equation of motion)

\[= - \frac{\partial}{\partial x} ( \frac{\partial p}{\partial x}) = \frac{\partial}{\partial x} (\frac{\rho \partial vx}{\partial t}) = \rho \frac{\partial}{\partial t} (\frac{\partial vx}{\partial x})\]

\[= \frac{-\partial^{2} p}{\partial x^{2}} = \rho \frac{\partial}{\partial t} (\frac{-1}{\frac{K \partial p}{\partial t}})\]

\[= \frac{\partial p^{2}}{\partial x^{2}} - \frac{\rho}{K} \frac{\partial^{2} p}{\partial t^{2}} = 0\]

Rewriting the above equation gives us:

\[\frac{\partial^{2} u(x, t)}{\partial x^{2}} \frac{1  \partial^{2} u(x, t)}{v^{2} \partial t^{2}}\]

Hooke's Law

When English scientist Robert Hooke was investigating springs and elasticity in the 19th century, he observed that numerous materials had a similar feature when the stress-strain connection was analyzed. The force required to stretch the material was proportional to the extension of the material in a linear area. This is known as Hooke’s Law. Within the elastic limit of a material, Hooke's law indicates that the strain is proportional to the applied stress. When elastic materials are stretched, the atoms and molecules deform until stress is applied, and then they return to their original state when the stress is removed. Hooke’s law is expressed as - 

F = –kx

F is the force, x is the extension length, and k is the proportionality constant, also known as the spring constant in N/m, in the equation.