How do you derive a wave equation?

The wave equation is derived by applying F=ma to an infinitesimal length dx of string (see the diagram below). We picture our little length of string as bobbing up and down in simple harmonic motion, which we can verify by finding the net force on it as follows.

How do you solve a one dimensional wave equation?

Therefore, the general solution to the one dimensional wave equation (21.1) can be written in the form u(x, t) = F(x − ct) + G(x + ct) (21.6) provided F and G are sufficiently differentiable functions.

How many conditions are needed for one dimensional wave equation?

There are four boundary conditions. We apply them in turn. As with the wave equation there are easy orders in which to apply the boundary conditions, and orders which still work but are less easy.

Which method is used to find the solution of wave equation?

variational iteration method
An analytic approximation to the solution of wave equation is studied. Wave equation is in radial form with indicated initial and boundary conditions, by variational iteration method it has been used to derive this approximation and some examples are presented to show the simplicity and efficiency of the method.

What is a one-dimensional wave?

In the most general sense, waves are particles or other media with wavelike properties and structure (presence of crests and troughs). Wave Fundamentals. The simplest wave is the (spatially) one-dimensional sine wave (or harmonic wave or sinusoid) with an amplitude u described by the equation: u(x,t)=Asin(kx−ωt+ϕ)

Who discovered the one-dimensional wave equation?

Rond d’Alembert
French scientist Jean-Baptiste le Rond d’Alembert discovered the wave equation in one space dimension.

What are the assumptions in deriving one-dimensional wave equation?

In deriving this equation we make the following assumptions. (i) The motion takes place entirely in one plane i.e., XY plane. particles of the string is negligible. (iii)The tension T is constant at all times and at all points of the deflected string.

Who discovered the one-dimensional wave equation Mcq?

d’Alembert
In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.

What is a one-dimensional equation?

The Wave Equation 3 is called the classical wave equation in one dimension and is a linear partial differential equation. It tells us how the displacement u can change as a function of position and time and the function. The solutions to the wave equation (u(x,t)) are obtained by appropriate integration techniques.

How many possible solutions are there in one-dimensional wave equation?

Existence is clear: we exhibited a formula for the general solution, namely, (7.26). Unique- ness is also clear: there is only one solution defined by the initial data.

When solving a 1-Dimensional wave equation using the variable separable method we get the solution if AK is positive BK is negative CK is anything?

Explanation: Since the given problem is 1-Dimensional wave equation, the solution should be periodic in nature. If k is a positive number, then the solution comes out to be (c7 epx⁄c+e-px⁄cc8)(c7 ept+e-ptc8) and if k is positive the solution comes out to be (ccos(px/c) + c’sin(px/c))(c”cospt + c”’sinpt).

When solving 1 dimensional heat What is the equation?

Goal: Model heat (thermal energy) flow in a one-dimensional object (thin rod). u(x,t) = temperature in rod at position x, time t. ∂u ∂t = c2 ∂2u ∂x2 . (the one-dimensional heat equation ) The constant c2 is called the thermal difiusivity of the rod.

How do you derive the wave equation in one dimensional case?

CHAPTER V WAVE THEORY 5.1 DERIVATION OF ONE DIMENSIONAL WAVE EQUATION. The wave equation in the one dimensional case can be derived from Hooke’s law in the following way: Imagine an array of little weights of mass. m. are interconnected with mass less springs of length. h. and the springs have a stiffness of. k. .

When was the wave equation discovered?

The wave equation arises in fields like fluid dynamics, electromagnetics, and acoustics. d’Alembert discovered the one-dimensional wave equation in the year 1746, after ten years Euler discovered the three-dimensional wave equation.

How can the wave equation be derived from Hooke’s law?

The wave equation in the one dimensional case can be derived from Hooke’s law in the following way: Imagine an array of little weights of mass m are interconnected with mass less springs of length h and the springs have a stiffness of k Here ux() measures the distance from the equilibrium of the mass situated at x