Chapter 16: Q. 64 (page 453)

Is the displacement $D (x, t) = (0.10 - 0.10 x^{2} + x t - 2.5 t^{2}) m$ ,
where x is in m and t is in s, a possible traveling wave? If so, what
is the wave speed?

Short Answer

Expert verified

The given function is a possible travelling wave and it has a wave speed of $5 m / s .$

Step by step solution

Given information

Any function can be considered as a travelling wave if it satisfies the wave equation,

$\frac{\partial^{2} f}{\partial t^{2}} = v^{2} \frac{\partial^{2} f}{\partial x^{2}}$

Therefore, we have to verify whether the function $D (x, t) = (0.10 - 0.10 x^{2} + x t - 2.5 t^{2}) m$ ,satisfies the wave equation.

Verification of the function

The given function can be written as,

$D (x, t) = (0.10 - 0.10 x^{2} + x t - 2.5 t^{2}) m D (x, t) = 0.10 [1 - {(x - 5 t)}^{2}] m$

Therefore, taking the partial derivative we get,

$\frac{\partial D}{\partial t} = 0.10 \times (- 2) (x - 5 t) (- 5) m / s \Rightarrow \frac{\partial D}{\partial t} = (x - 5 t) m / s \Rightarrow \frac{\partial^{2} D}{\partial t^{2}} = - 5 m / s^{2}$

Also, taking partial derivative w.r.t $' x',$ we get,

$\frac{\partial D}{\partial x} = 0.10 \times (- 2) (x - 5 t) (1) \Rightarrow \frac{\partial D}{\partial x} = - 0.2 \times (x - 5 t) m^{- 1} \Rightarrow \frac{\partial^{2} D}{\partial x^{2}} = - 0.2 m^{- 1}$

Therefore, substituting the values obtained in the wave equation, we get,

$\frac{\partial^{2} D}{\partial t^{2}} = v^{2} \frac{\partial^{2} D}{\partial x^{2}} \Rightarrow - 5 m / s^{2} = v^{2} (- 0.2 m^{- 1}) \Rightarrow v^{2} = 25 m^{2} / s^{2} \Rightarrow v = 5 m / s$

Thus, the given function is a possible travelling wave and it has a wave speed of $5 m / s .$

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Is the displacement $D (x, t) = (0.10 - 0.10 x^{2} + x t - 2.5 t^{2}) m$ ,
where x is in m and t is in s, a possible traveling wave? If so, what
is the wave speed?

Short Answer

Step by step solution

Given information

Verification of the function

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Is the displacement Dx,t=0.10-0.10x2+xt-2.5t2m,where x is in m and t is in s, a possible traveling wave? If so, whatis the wave speed?

Short Answer

Step by step solution

Given information

Verification of the function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Electricity

Quantum Physics

Classical Mechanics

Translational Dynamics

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.

Is the displacement $D (x, t) = (0.10 - 0.10 x^{2} + x t - 2.5 t^{2}) m$ ,
where x is in m and t is in s, a possible traveling wave? If so, what
is the wave speed?