Problem 22 - Physics

Question

The displacement (in meters) of a wave is given according to \(y=0.26 \sin (\pi t-3.7 \pi x)\) where \(t\) is in seconds and \(x\) is in meters, (a) Is the wave traveling in the \(+x\) or \(-x\) direction? (b) What is the displacement \(y\) when \(t=38 \mathrm{~s}\) and \(x=13 \mathrm{~m} ?\)

Step-by-Step Solution

Answer

(a) The wave is traveling in the \(-x\) direction. (b) The displacement is approximately \(-0.08034\) meters.

Step 1: Identify Wave Direction

The wave function is given by \( y = 0.26 \sin(\pi t - 3.7\pi x) \). A basic wave equation is of the form \( y = A \sin(kx - \omega t) \) for a wave traveling in the \(+x\) direction and \( y = A \sin(\omega t - kx) \) for a wave traveling in the \(-x\) direction. Since our equation matches the form \( y = A \sin(\omega t - kx) \), the wave is traveling in the \(-x\) direction.

Step 2: Plug in Values for Displacement

To find the displacement \( y \) at \( t = 38 \) s and \( x = 13 \) m, substitute these values into the wave equation: \[ y = 0.26 \sin(\pi \times 38 - 3.7 \pi \times 13) \] which simplifies to \[ y = 0.26 \sin(38\pi - 48.1\pi) \].

Step 3: Simplify the Argument of Sine

Simplify the expression inside the sine function: \[ 38\pi - 48.1\pi = -10.1\pi \]. Now the equation becomes \( y = 0.26 \sin(-10.1\pi) \).

Step 4: Evaluate the Sine Function

The sine function is periodic with a period of \(2\pi\), thus \( \sin(-10.1\pi) = \sin(-10\pi - 0.1\pi) \). Since \(\sin(-10\pi) = 0\), what remains is \( \sin(-0.1\pi) \). As \( -0.1\pi \) is an angle where the sine function is negative, calculate it precisely: \( \sin(-0.1\pi) \approx -0.309 \).

Step 5: Compute the Displacement

Now calculate \( y \) using the value from the sine function: \( y = 0.26 \times -0.309 \approx -0.08034 \) meters. Therefore, the displacement at \( t = 38 \) s and \( x = 13 \) m is approximately \(-0.08034\) meters.

Key Concepts

Wave Equation

When studying waves, the wave equation is a fundamental concept that describes the behavior of waves. In general, a wave can be described by the equation:

• For a wave traveling to the right (positive x-direction): \[ y = A \sin(kx - \omega t) \]
• For a wave traveling to the left (negative x-direction): \[ y = A \sin(\omega t - kx) \]

Here,

• \(y\) is the wave displacement.
• \(A\) is the amplitude, indicating how high the wave gets.
• \(kx\) is the wave number term, showing the wave's spatial periodicity.
• \(\omega\) is the angular frequency, which reveals how fast the wave oscillates in time.

The signs in the equation help to determine the direction of wave motion, which is a crucial element when analyzing wave problems.

Wave Direction

Determining the wave direction is essential for understanding wave behavior. The direction of the wave shows whether the wave is traveling towards the positive or negative direction along the x-axis. When faced with a wave equation like \( y = 0.26 \sin(\pi t - 3.7\pi x) \), recognize the format:

• The wave travels in the negative x-direction if it's in the form \( y = A \sin(\omega t - kx) \).
• Conversely, it travels in the positive x-direction if it's \( y = A \sin(kx - \omega t) \).

In the given exercise, the wave's format, \( y = 0.26 \sin(\pi t - 3.7\pi x) \), follows the \( y = A \sin(\omega t - kx) \) structure, signaling that the wave indeed travels in the \(-x\) direction.

Sine Function

The sine function is integral when analyzing wave displacements. In a wave equation like the one we have here, \( y = 0.26 \sin(\pi t - 3.7\pi x) \), the sine function is responsible for the oscillatory nature of the wave.

• The argument of the sine function — \(\pi t - 3.7\pi x\) — determines the phase and ultimately how the wave oscillates.
• Understanding how this argument changes helps us determine specific values, such as the displacement of the wave.

The properties of the sine function, including its periodic nature with a period of \(2\pi\), are critical. This periodicity explains why the sine function’s value repeats over given intervals. Knowing that it passes through zero at multiples of \(\pi\) and peaks at \(\pi/2 + 2n\pi\) for odd integers \(n\), helps in evaluating wave problems.

Periodic Functions

Periodic functions repeat their values in regular intervals, a characteristic crucial to understanding waves. In our wave equation, the sine function is periodic, meaning:

• It repeats every \(2\pi\).
• This regularity ensures that wave patterns are predictable over time and space.

These periodic properties allow us to predict future wave behavior based on past observations. For instance, by evaluating the sine function, we determined the displacement at specific time and position values. Recognizing the pattern in periodic functions allows for calculating both predictable and complex wave phenomena, even without visual observation.