Show that the displacement D(x, t) = ln(ax + bt) , where a and b are constants, is a solution to the wave equation. Then find an expression in terms of a and b for the wave speed.
The given displacement is a solution to the wave equation.
The wave speed is b/a .
The displacement is given by where a, b are constants
The given displacement must satisfy the wave equation .
Take the partial derivatives of D(x, t) with respect to x,
Take the partial derivatives of D(x, t) with respect to t,
Substitute the values in the above wave equation:
Therefore, the given displacement satisfies the wave equation.
As calculated in the above equation, the wave speed is obtained as .
A sound source is located somewhere along the x-axis. Experiments show that the same wave front simultaneously reaches listeners at x = -7.0 m and x = +3.0 m. a. What is the x-coordinate of the source? b. A third listener is positioned along the positive y-axis. What is her y-coordinate if the same wave front reaches her at the same instant it does the first two listeners?
You have just been pulled over for running a red light, and
the police officer has informed you that the fine will be $250. In
desperation, you suddenly recall an idea that your physics professor recently discussed in class. In your calmest voice, you tell the officer that the laws of physics prevented you from knowing that the light was red. In fact, as you drove toward it, the light was Doppler shifted to where it appeared green to you. “OK,” says the officer, “Then I’ll ticket you for speeding. The fine is $1 for every 1 km/h over the posted speed limit of 50 km/h.< How big is your fine? Use 650 nm as the wavelength of red light and 540 nm as the wavelength of green light.
A distant star system is discovered in which a planet with
twice the radius of the earth and rotating 3.0 times as fast as the
earth orbits a star with a total power output of .
a. If the star’s radius is 6.0 times that of the sun, what is the
electromagnetic wave intensity at the surface? Astronomers
call this the surface flux. Astronomical data are provided
inside the back cover of the book.
b. Every planet-day (one rotation), the planet receives .
of energy. What is the planet’s distance from its star? Give
your answer in astronomical units (AU), where 1 AU is the
distance of the earth from the sun.
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