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(II) Show by direct substitution that the following functions satisfy the wave equation: $(a) \quad D(x, t)=A \ln (x+v t)$(b) $D(x, t)=(x-v t)^{4}$

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a) Yesb) Yes

03:29

Kai Chen

Physics 101 Mechanics

Physics 102 Electricity and Magnetism

Chapter 15

Wave Motion

Periodic Motion

Mechanical Waves

Electromagnetic Waves

Cornell University

University of Michigan - Ann Arbor

Simon Fraser University

Lectures

03:40

In physics, electromagnetic radiation (EM radiation or EMR) refers to the waves (or their quanta, photons) of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays. Electromagnetic waves of different frequency are called by different names since they have different sources and effects on matter. In order of increasing frequency and decreasing wavelength these are: radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays.

10:59

In physics, Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. They underpin all electric, optical and radio such electromagnetic technologies as power generation, electric motors, wireless communication, cameras, televisions, computers, and radar. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of these fields. The equations have two major variants. The microscopic Maxwell equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the atomic scale. The macroscopic Maxwell equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic scale details. The equations were published by Maxwell in his 1864 paper "A Dynamical Theory of the Electromagnetic Field". In the original paper Maxwell fully derived them from the Lorentz force law (without using the Lorentz transformation) and also from the conservation of energy and momentum.

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Show that the wave functio…

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Show that the functions ar…

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Show that the wave functi…

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Show that the function $D=…

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(a) Show that the function…

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Show that the following fu…

for this problem on the topic off wave motion, we want to use direct substitution to show that to functions that are given satisfy the wave equation not to be a solution off the way the creation, the function must satisfy the differential equation. D two d d X squared partial D two d d X squared is able to one over the speed of the wave v squared times partial d two d d t squared. So we have to find the second derivatives with respect to position and time for each function. So in part A, we have the function D as a function of X anti is equal to a times the natural log off X plus VT. So let's just take the derivative with respect to X. So the partial derivative D D. X is equal to a over X plus V t. And then from here, we can take the We can take the derivative of this with respect to X, which gives us the second revenue off D with respect to x d two x d. Two d, The X squared two B minus A divided by ex U. S. V. T or squared Now we need to do the same for the function with respect to t So partial d d d t is equal to a times we over x less VT On the second derivative off the with respect to t d two d d t squared is able to a V squared minus a B squared over X plus V t on square. So if we substitute this directly into the wave equation above, we find that d two d. The X squared is equal to one over v squared d two d, he t script. So again, yes, we can see that this function by the next substitution does satisfy the way the creation now in part B were given a different function. But we will perform the same procedure. So in part, we were given the function d off X and T is equal to X minus V t or to the power form. So let's find the first derivative with respect to X So partial d d. The X is he put you full into X minus VT or cute and then the second derivative with respect to X D to a the X squared is equal to 12 into X minus V T or square. Now we'll take the derivative of this function with respect to T So D d DT is equal to minus for the into x minus VT or cute and D two d d t squared. The second derivative with respect to time is equal to 12 v squared into X minus VT or square. So, again, by the that substitution, we get that the second derivative of this function with respect two x d two d d X squared. Is he going to one over v squared, multiplied by the second derivative of this function with respect to time d two d, he t squared. So again, yes, this function is a solution off the rave in creation.

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