Downloaded 92 times
Uploaded byMuhammad Saqib
The wave eqution presentation
The document discusses the wave equation. It begins by defining the 1D wave equation and extending it to 2D. It then derives the wave equation for an ideal string by considering tension, mass per unit length, and applying Newton's second law. Taking derivatives, the document arrives at the wave equation and shows its solution describes waves moving left and right. It concludes that the most general solution to the wave equation is the sum of left- and right-moving waves, making it linear.
More Related Content
Similar to The wave eqution presentation
The wave eqution presentation
- 2. Applied Physics Presentation The WaveEquation Muhammad
- 3. What is theWave Equation? The wave equation for a plane wave traveling in the x direction is: (A) where v is the phase velocity of the wave and y represents the variable which is changing as the wave passes.
- 4. Wave Equation in2-Dimensions The mathematical description of a wave makes use of partial derivatives. In two dimensions the wave Equation takes the form (B)
- 5. The Wave inan Ideal String We start with a small portion of that rope. We have Tension ‘T’ and mass per unit length ‘µ’. If our displacement is not absurdly high then the Tension is the same on the both side.
- 6. Derivation of theWave Equation We will only consider the motion in the Y direction. So That’s for small angles other wise all our assumptions will be wrong. When So now the Equation becomes: (1)
- 7. Appling Newton's SecondLaw The amount of mass that is in Since, here is From (1), (2)
- 8. What is ? We knowthat since the length of So, Where length. is is the amount of mass per unit .
- 9. So the Equation(2) becomes, (4)
- 10. Since in alimiting case we are going to make , We will find the tangent for θ, (i) We know that tangent of “theta” is always equal to the derivative in space (position). We used the partial derivative since we assume that its all happening on a instant in time.
- 11. Taking Derivative onboth sides of (i), we get (ii) For Small Angle Approximation,
- 12. Thus the Equation(ii) becomes… (iii)
- 13. Substituting in equation (4). Weget, (5)
- 14. The Solution tothe differential Equation (5) Where c is the constant. We know that the dimensions of ct are the same as that of x. Thus,
- 15. Above function willsatisfy the differential equation (5). When we take the 2nd derivative in time we get the out and we get the 2nd derivative of the function. Take the 2nd derivative in x and we only get the 2nd derivative of the function.
- 16. The only thingrequired is… The value of ‘c’ The dimensions of c is meter per seconds i.e. c is the velocity.
- 17. Thus we changeour equation into a more uniform way This is the Equation that is generally called the wave Equation.
- 18. We know thatx is the displacement of wave along x axis. is a point back in time at t=0 and displacement x’=x-ct
- 19. The Other Solution… SinceThe Wave Equation Involves a square of v, So we can generate another class of solutions by simply changing the sign of velocity i.e.
- 20. Most surprising Result… Themost general solution to the wave Equation is the sum of a wave to the left and a wave to the right. The most general solution Thus wave Equation is Linear since the sum of two solutions is itself a solution.
- 21. ? How the wavetravels in a string Suppose that two persons are holding a rope of constant thickness. The person the right side jerks the rope and a wave forms in the rope. While travelling towards the guy on left It travels, let’s say, as a Mountain. While On its way back it’s a valley.
- 22. A Why is itso? B We know that the point B is Fixed so when the wave passes through point B then to have zero displacement point B moves exactly the same distance downwards. Thus producing a valley as a result of a mountain.