circular, is rectangular with length a and width b. Find the general solutions for waves on the membrane stretched over the rectangle when the initial hit is given by g(x, y). The wave equation for this situation is dy? c² Ət? where c is the speed of the waves on the membrane. Choose a coordi- nate system where x = 0, y = 0 is at one corner of the drum and it is priented with the length a along x and the length b along y.
circular, is rectangular with length a and width b. Find the general solutions for waves on the membrane stretched over the rectangle when the initial hit is given by g(x, y). The wave equation for this situation is dy? c² Ət? where c is the speed of the waves on the membrane. Choose a coordi- nate system where x = 0, y = 0 is at one corner of the drum and it is priented with the length a along x and the length b along y.
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![**Wave Equation on a Rectangular Drum**
A musician has constructed an unusual drum that, instead of being circular, is rectangular with length \(a\) and width \(b\). The task is to find the general solutions for waves on the membrane stretched over the rectangle when the initial hit is given by \(g(x, y)\).
The wave equation for this situation is:
\[
\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2}
\]
where \(c\) is the speed of the waves on the membrane.
Choose a coordinate system where \(x = 0\), \(y = 0\) is at one corner of the drum. The coordinate system is oriented with the length \(a\) along the \(x\)-axis and the length \(b\) along the \(y\)-axis.
This equation is a partial differential equation representing the behavior of waves on a rectangular membrane. It requires determining the function \(\phi(x, y, t)\) that satisfies the boundary conditions and initial conditions given by \(g(x, y)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1266795a-ae85-4a96-b8ce-46e21249f320%2F821a58ea-58d5-488f-9c15-ad1c09b8c579%2Fewsbopl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Wave Equation on a Rectangular Drum**
A musician has constructed an unusual drum that, instead of being circular, is rectangular with length \(a\) and width \(b\). The task is to find the general solutions for waves on the membrane stretched over the rectangle when the initial hit is given by \(g(x, y)\).
The wave equation for this situation is:
\[
\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2}
\]
where \(c\) is the speed of the waves on the membrane.
Choose a coordinate system where \(x = 0\), \(y = 0\) is at one corner of the drum. The coordinate system is oriented with the length \(a\) along the \(x\)-axis and the length \(b\) along the \(y\)-axis.
This equation is a partial differential equation representing the behavior of waves on a rectangular membrane. It requires determining the function \(\phi(x, y, t)\) that satisfies the boundary conditions and initial conditions given by \(g(x, y)\).
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