Wave Equation and Standing Waves. (a) Prove by direct substitution that y(x, t)=(A sw sink x) s

Wave Equation and Standing Waves. (a) Prove by direct...

Key Concepts

Wave Speed Relation (Dispersion Relation)

The wave speed relation, expressed as v = ?/k, connects the angular frequency ? and the wavenumber k to the speed at which the wave propagates. This relation holds for both traveling and standing waves because even though standing waves do not transport energy through space, their underlying components are traveling waves that obey this fundamental dispersion relationship.

Wave Equation

The wave equation is a second-order partial differential equation that models how wave phenomena propagate through a medium. It relates the second spatial derivative of the wave function to its second time derivative, capturing the balance between spatial curvature and temporal acceleration in the wave's evolution.

Standing Waves

Standing waves are formed by the superposition of two waves traveling in opposite directions, resulting in a fixed pattern of nodes and antinodes where the amplitude remains constant in space over time. This phenomenon arises in systems with reflective boundaries or resonant conditions, allowing the wave pattern to appear stationary.

Direct Substitution Method

The method of direct substitution involves differentiating a proposed solution with respect to its spatial and temporal variables and then substituting these derivatives into the original wave equation. If the resulting expression satisfies the equation identically, the solution is verified and the parameters, such as the wave speed, are confirmed.

Transcript

00:01 So to obtain our probe that we need for this problem, we need to start from the general wave equation, which is written exactly here.

00:12 And if the given function, which is the standing wave function, y equals a sinus kx, sine as omega t, so we need to use this, this function and we need to find all of these derivatives all of this second partial derivatives so this is one second partial derivative and this is one second partial derivative or the partial derivatives of the second order so we need to find these derivatives and if if this equation stands true so if we have on the left hand side the second order partial derivative of the y function over x and then if that expression exactly equals 1 over velocity squared, which is given here in this form, times the second partial derivative of y over t, if that that equality stands exactly, then this function y is the proper solution for this differential equation and is a proper wave function.

01:33 So we will immediately start from finding the second derivative over x.

01:39 So this is first, we will find this first partial derivative over x, which equals here we derive only from the sign of kx.

01:50 So we will have the derivative from the sine is cosine, and the derivative from the nest function in sine the sign, which is kx is only k.

02:00 So we'll have k times cosine.

02:04 Cosine kx times sine omega t and then the second derivative is the derivative of the first derivative so from this from this here so this will be again only derivative of the cosine and the derivative of the cosine is negative sign and the derivative of the inside function kx is again k so this will be negative k squared sine kx, sine omega -t.

02:39 And in the same manner, we will define derivatives for the time.

02:43 So in the case of the time, we will only derive from the sinus omega -t part.

02:48 So this will be omega -goes in front, times sine kx.

02:54 We don't touch this part, and then cosine of omega -t.

02:59 And for the second derivative, second partial derivative, again we will derive from the which will be negative sign and again from the inside function omega product t which will be only omega so this will be negative omega squared sine kx sine x sine omega t and let's see what happens if we substitute this into the wave equation so on one hand side we need to on the left hand side we need to substitute the the x and the derivative over x so this is going to be negative k squared sin is k x sinus omega t in that equals 1 over v squared negative omega squared sinus k x sinus omega t what we can see here we need to substitute the relationship between the velocity and the wave number and the angular frequency.

04:28 So we will do that immediately to see if we get exact equality.

04:36 I will rewrite this again...

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