Concept explainers
In Section 16.7, we derived the speed of sound in a gas using the impulse–momentum theorem applied to the cylinder of gas in Figure 16.20. Let us find the speed of sound in a gas using a different approach based on the element of gas in Figure 16.18. Proceed as follows. (a) Draw a force diagram for this element showing the forces exerted on the left and right surfaces due to the pressure of the gas on either side of the element. (b) By applying Newton’s second law to the element, show that
(c) By substituting ΔP = −(B ∂s/∂x) (Eq. 16.30), derive the following wave equation for sound:
(d) To a mathematical physicist, this equation demonstrates the existence of sound waves and determines their speed. As a physics student, you must take another step or two. Substitute into the wave equation the trial solution s(x, t) = smax cos (kx − ωt). Show that this function satisfies the wave equation, provided
(a)
The force diagram for this element showing the force exerted on the left and the right surface.
Answer to Problem 17.72CP
The force diagram for this element showing the force exerted on the left and the right surface is
Explanation of Solution
Force diagram contains all the forces acting on the body. It contains the direction of the each force acting on the body represents at its top and bottom end or left and right sides.
The force diagram for this element showing the force exerted on the left and the right surface is shown below.
Figure (1)
The force diagram of the element of gas in Figure (1) indicates the force exerted on the right and left surfaces due the pressure of the gas on the either side of the gas.
(b)
The expression,
Answer to Problem 17.72CP
The expression
Explanation of Solution
Let
The net force to the right on the chunk of air in Figure (1) is,
The force due to atmosphere is,
Here,
Differentiate the equation (1) with respect to
Formula to calculate the mass of the air is,
Here,
Formula to calculate the acceleration is,
Here,
From Newton’s second law, formula to calculate the Force is,
Substitute
Conclusion:
Therefore the expression,
(c)
The wave equation for sound is
Answer to Problem 17.72CP
The following wave equation for sound is
Explanation of Solution
The value of the
From part (b), the given expression is,
Substitute
Thus, the wave equation for sound is
Conclusion:
Therefore, the wave equation for sound is
(d)
The function
Answer to Problem 17.72CP
The function
Explanation of Solution
The given wave equation is,
Apply the trial solution in the above equation.
Double differentiate the equation (1) with respect to
Double differentiate the equation (1) with respect to
The wave equation for sound in part (c) is,
Substitute
Thus, the function
Conclusion:
Therefore, the function
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