/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 174 The \((x, y)\) co-ordinates of t... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 174

The \((x, y)\) co-ordinates of the corners of a square plate are \((0,0),(L, 0),(L, L)\), and \((0, L)\). The edges of the plate are clamped and transverse standing waves are set up in it. If \(u(x, y)\) denotes the displacement of the plate at the point \((x, y)\) at some instant of time, the possible expression(s) for \(u\) is (are) ( \(a=\) positive constant) (A) \(a \cos (\pi x / 2 L) \cos (\pi y / 2 L)\) (B) \(a \sin (\pi x / L) \sin (\pi y / L)\) (C) \(a \sin (\pi x / L) \sin (2 \pi y / L)\) (D) \(a \cos (2 \pi x / L) \sin (\pi y / L)\)

Short Answer

Expert verified
The possible expression(s) for \(u(x, y)\) are: A) \(a \cos (\frac{\pi x}{2L}) \cos (\frac{\pi y}{2L})\) B) \(a \sin (\frac{\pi x}{L}) \sin (\frac{\pi y}{L})\)

Step by step solution

01

Identify Required Conditions

Since the edges of the plate are clamped, we must have \(u(x,y) = 0\) on the edges. This means that the expression for \(u(x,y)\) must be zero when \(x=0\), \(x=L\), \(y=0\), or \(y=L\). Let's analyze each given expression and see if it meets this condition.
02

Analyze Option A

Option A: \(a \cos(\frac{\pi x}{2L}) \cos(\frac{\pi y}{2L})\). In this expression: When \(x=0\), \(\cos(\frac{\pi x}{2L}) = \cos(0) = 1\); similar when \(y=0\), \(\cos(\frac{\pi y}{2L}) = 1\). When \(x=L\), \(\cos(\frac{\pi x}{2L}) = \cos(\frac{\pi L}{2L}) = \cos(\frac{\pi}{2}) = 0\). Also, when \(y=L\), \(\cos(\frac{\pi y}{2L}) = \cos(\frac{\pi L}{2L}) = \cos(\frac{\pi}{2}) = 0\). Therefore, the expression is zero on all the edges as desired, so Option A is a possible expression for \(u(x,y)\).
03

Analyze Option B

Option B: \(a \sin(\frac{\pi x}{L}) \sin(\frac{\pi y}{L})\). In this expression: When \(x=0\), \(\sin(\frac{\pi x}{L}) = \sin(0) = 0\); similar when \(y=0\), \(\sin(\frac{\pi y}{L}) = 0\). When \(x=L\), \(\sin(\frac{\pi x}{L}) = \sin(\pi) = 0\). And when \(y=L\), \(\sin(\frac{\pi y}{L}) = \sin(\pi) = 0\). Therefore, the expression is zero on all the edges as desired, so Option B is a possible expression for \(u(x,y)\).
04

Analyze Option C

Option C: \(a \sin(\frac{\pi x}{L}) \sin(\frac{2\pi y}{L})\). In this expression: When \(x=0\), \(\sin(\frac{\pi x}{L}) = \sin(0) = 0\), satisfying this condition. Also when \(x=L\), \(\sin(\frac{\pi x}{L}) = \sin(\pi) = 0\). When \(y=0\), \(\sin(\frac{2\pi y}{L}) = \sin(0) = 0\). But when \(y=L\), \(\sin(\frac{2\pi y}{L}) = \sin(2\pi) = 0\), which is not true. Therefore, Option C doesn't satisfy the condition for all edges and is not a possible expression for \(u(x,y)\).
05

Analyze Option D

Option D: \(a \cos(\frac{2\pi x}{L}) \sin(\frac{\pi y}{L})\). In this expression: When \(x=0\), \(\cos(\frac{2\pi x}{L}) = \cos(0) = 1\) and when \(x=L\), \(\cos(\frac{2\pi x}{L}) = \cos(2\pi) = 1\), satisfying this condition for \(x\). When \(y=0\), \(\sin(\frac{\pi y}{L}) = \sin(0) = 0\). But when \(y=L\), \(\sin(\frac{\pi y}{L}) = \sin(\pi) = 0\), which is not true. Therefore, Option D doesn't satisfy the condition for all the edges and is not a possible expression for \(u(x,y)\). #Conclusion#: From our analysis, the possible expression(s) for \(u(x, y)\) are: A) \(a \cos (\frac{\pi x}{2L}) \cos (\frac{\pi y}{2L})\) B) \(a \sin (\frac{\pi x}{L}) \sin (\frac{\pi y}{L})\)

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Most popular questions from this chapter

A rope, under tension of \(200 \mathrm{~N}\) and fixed at both ends, oscillates in a second harmonic standing wave pattern. The displacement of the rope is given by \(y=(0.10 \mathrm{~m}) \sin \left(\frac{\pi x}{2}\right) \sin (12 \pi t)\), where \(x=0\) at one end of the rope, \(x\) is in metres, and \(t\) is in seconds. Find (A) the length of the rope (B) the speed of waves on the rope (C) the mass of the rope (D) if the rope oscillates in a third harmonic standing wave pattern, what will be the period of oscillation?

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