Calc1231W3T3 - The Wave Equation

Saturday, 4 August 2018

9:23 PM

Untitled picture.png Machine generated alternative text: Given a function, it is relatively easy to find the partial derivatives. For example suppose u(x, t) Dt We say this function u satisfies, or is a solution to, the differential equation sin(x + ct) then Du Dt Du Similarly if t'(x, t) = sin@ — ct) then Dv Dt So this function v satisfies the differential equation Dv D-v Dt DC Untitled picture.png Machine generated alternative text: Suppose u@, t) = sin@ + ct) where c is a constant. There are four possible derivatives of order two. These are 32 u Dt2 D2u DtDx D2u 32 u Dx2 So this function satisfies a very famous partial differential equation, which originated from the study of a vibrating string on a musical instrument, called the one dimensional wave equation. 32 u c2 D2u Dt2 Dx2 The function v(x, t) = sin@ — ct) is also a solution to this equatiom The functions u(x, t), v(x, t) and u@,t) + v(x, t) are plotted in red, orange and purple in the picture below with c=l: Assuming c > O then the red function sin@ + ct) is a wave that moves to the lett O over time; while the orange function sin(x ct) is a wave that moves to the right O over time The sum ofthe left and right moving waves is the standing wave sin@ + ct) + sin@ — ct) in purple.
Untitled picture.png Machine generated alternative text: Suppose u@, t) = sin@ + ct) where c is a constant. There are four possible derivatives of order two. These are 32 u Dt2 D2u DtDx D2u 32 u Dx2 So this function satisfies a very famous partial differential equation, which originated from the study of a vibrating string on a musical instrument, called the one dimensional wave equation. 32 u c2 D2u Dt2 Dx2 The function v(x, t) = sin@ — ct) is also a solution to this equatiom The functions u(x, t), v(x, t) and u@,t) + v(x, t) are plotted in red, orange and purple in the picture below with c=l: Assuming c > O then the red function sin@ + ct) is a wave that moves to the lett O over time; while the orange function sin(x ct) is a wave that moves to the right O over time The sum ofthe left and right moving waves is the standing wave sin@ + ct) + sin@ — ct) in purple. Untitled picture.png Machine generated alternative text: The function u@, t) ei(x+ct) is another solution to the wave equation D2u Dt2 c2 D2u Let's investiage the case when c = 2 + 4i _ Then • The real part of u is • The imaginary part of u is o fit So the function oscillates in the complex plane; but these oscillations are dampened by the exponential term. As time goes on lim u(x,t) Untitled picture.png Machine generated alternative text: D2u Dt2 D2u -cos(x+ct) Dx2 02 v Dt2 02 v -cos(x-ct) Dx2 These functions are both solutions to the one-dimensional wave equation because 32 u Dt2 and D2t' Dt2 Consider u(x, t) = cos(x + ct) and v(x, t) —o —1 cos@ ct.). Then c2 D2u DC 2 c2 D2v Dx2 The functions u@,t),t'(x, t) and t) + t'(x, t) are plotted in red, orange and purple below with c 311/2 Ink Drawings Ink Drawings Ink Drawings           
Untitled picture.png Machine generated alternative text: D2u Dt2 D2u -cos(x+ct) Dx2 02 v Dt2 02 v -cos(x-ct) Dx2 These functions are both solutions to the one-dimensional wave equation because 32 u Dt2 and D2t' Dt2 Consider u(x, t) = cos(x + ct) and v(x, t) —o —1 cos@ ct.). Then c2 D2u DC 2 c2 D2v Dx2 The functions u@,t),t'(x, t) and t) + t'(x, t) are plotted in red, orange and purple below with c 311/2 Untitled picture.png Machine generated alternative text: We have seen that the one-dimensional wave equation has lots of possible solutions In fact there is a whole vector subspace of solutions (you may like to prove this). We can narrow down the range of possible solutions by adding extra conditions that the function must satisfy. These are usually called initial conditions. For example, suppose that u@, t) satisfies the one-dimensional wave equation: D 2 u c2 D2u Dt2 Dx2 In addition, suppose u@, t) satisfies the initial conditions: Du O) = and In 1746; Jean-Baptiste le Rond d'Alembert discovered that there is only one possible solution which satisfies these conditions: u(x, t) — 1 1 2 g(s)ds. x—ct Jean-Baptise le Rond d'Alembert (1717 -1783) For example, if we have the wave equation with c and initial conditions f@) = sin(x) and g@) — cos@) then x —ct So by d'Alembert's formula, the solution is g(s)ds 02 u Dt2 cos(s)ds 02 u Dx2 sin(x+t) - sin(x-t)
Untitled picture.png Machine generated alternative text: We have seen that the one-dimensional wave equation has lots of possible solutions In fact there is a whole vector subspace of solutions (you may like to prove this). We can narrow down the range of possible solutions by adding extra conditions that the function must satisfy. These are usually called initial conditions. For example, suppose that u@, t) satisfies the one-dimensional wave equation: D 2 u c2 D2u Dt2 Dx2 In addition, suppose u@, t) satisfies the initial conditions: Du O) = and In 1746; Jean-Baptiste le Rond d'Alembert discovered that there is only one possible solution which satisfies these conditions: u(x, t) — 1 1 2 g(s)ds. x—ct Jean-Baptise le Rond d'Alembert (1717 -1783) For example, if we have the wave equation with c and initial conditions f@) = sin(x) and g@) — cos@) then x —ct So by d'Alembert's formula, the solution is g(s)ds 02 u Dt2 cos(s)ds 02 u Dx2 sin(x+t) - sin(x-t)

 

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