Solving Sin(theta) = dy/dx | Acoustics Course Prep

  • Thread starter rexregisanimi
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In summary, when θ is small, the sine function can be approximated by the tangent function, which is equivalent to the slope of the string in the derivation for the vibrating string.
  • #1
rexregisanimi
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In preparing for an acoustics course, I ran across the following sentence which confused me:

"If (theta) is small, sin(theta) may be replaced by [partial]dy/dx."

I expected to see sin(theta) = (theta) so this threw me off. This came up in the derevation of the one dimensional wave equation after approximating (by Taylor series) the transverse force on a mass element of a tensioned string with [partial]d(Tsin(theta))/dx. The approximation in question thus gave T*([partial]d2y/dx2)*dx.

In the original setup, x and y are cartesian axis in physical 2D space and (theta) is the angle the string (with tension T) makes from the x-axis after displacement from equalibrium.

I've never seen sine approximated by dy/dx before and was hoping somebody might shed some light for me :)
 
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  • #2
When ##\theta## is small, ##sin\theta \approx \tan\theta = \frac{dy}{dx}##. In the derivation for the vibrating string, ##\theta## is the slope angle of the string.
 
  • #3
Thank you! :)
 
  • #4
sin(θ) = Δy / sqrt(Δy^2 + Δx^2). For small θ, Δy is small compared to Δx, so

sin(θ) ≈ Δy / sqrt(0 + Δx^2) = Δy / Δx
 
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  • #5


I can provide some clarification on the use of sin(theta) and dy/dx in the context of acoustics and the derivation of the one-dimensional wave equation.

First, it is important to understand that in acoustics, the sine function is commonly used to describe the relationship between the displacement of a medium (such as a string or air) and its position. This is because the sine function is a periodic function, meaning that it repeats itself over a certain interval. In the case of a vibrating string, the displacement of the string can be described by a sine wave, with the position on the x-axis representing time and the amplitude representing the displacement.

Now, in the context of the one-dimensional wave equation, the partial derivative of y with respect to x (dy/dx) represents the rate of change of displacement along the x-axis. This is essentially the slope of the displacement curve at a given point. So, when we take the partial derivative of Tsin(theta) with respect to x, we are essentially calculating the slope of the sine wave at that point.

When (theta) is small, the sine function can be approximated by its first derivative, which is dy/dx. This is because when (theta) is small, the sine curve becomes very close to a straight line, and the slope of a straight line is equal to its first derivative.

So, in the context of the acoustics course, the sentence "If (theta) is small, sin(theta) may be replaced by [partial]dy/dx" is essentially saying that when (theta) is small, we can approximate the sine function with its first derivative, dy/dx. This is a common approximation used in physics and engineering when dealing with small angles.

I hope this helps to clarify the use of sine and dy/dx in the context of acoustics and the derivation of the one-dimensional wave equation.
 

1. What is the meaning of "Solving Sin(theta) = dy/dx" in the context of an Acoustics Course Prep?

"Solving Sin(theta) = dy/dx" refers to finding the value of theta (angle) that satisfies the equation, where theta represents the direction of a sound wave and dy/dx represents the rate of change of the sound wave's displacement over distance. In acoustics, this equation is often used to determine the direction of a sound source.

2. How is Sin(theta) related to sound waves?

Sin(theta) represents the sine function, which is used to describe the relationship between the angle of a sound wave and its displacement. In other words, it shows how the direction of a sound wave changes as it travels through a medium.

3. What is the importance of solving this equation in an Acoustics Course Prep?

Solving Sin(theta) = dy/dx is important in acoustics because it allows us to determine the direction of a sound wave, which can help us locate the source of the sound. This is particularly useful in fields such as noise control, architectural acoustics, and sonar technology.

4. What are some applications of solving Sin(theta) = dy/dx in acoustics?

Some applications of solving Sin(theta) = dy/dx in acoustics include noise mapping, sound source localization, and beamforming. It is also used in fields such as music production, concert hall design, and underwater acoustics.

5. What are some techniques for solving Sin(theta) = dy/dx in an Acoustics Course Prep?

There are various techniques for solving Sin(theta) = dy/dx, depending on the specific context and given information. Some common methods include using trigonometric identities, using vector calculus, and using numerical methods such as finite difference or finite element analysis. In an Acoustics Course Prep, students may also learn about specific applications and techniques, such as the beamforming method for sound source localization.

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