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 $5 m / s .$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electrostatics

Classical Mechanics

Work Energy and Power

Measurements

Further Mechanics and Thermal Physics

Fluids

Study anywhere. Anytime. Across all devices.

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

Electrostatics

Classical Mechanics

Work Energy and Power

Measurements

Further Mechanics and Thermal Physics

Fluids

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?