Problem 76 - Fundamentals of Physics

Question

A standing wave results from the sum of two transverse traveling waves given by $$ y_{1}=0.050 \cos (\pi x-4 \pi t) $$ and $$ y_{2}=0.050 \cos (\pi x+4 \pi t) $$ where \(x, y_{1}\), and \(y_{2}\) are in meters and \(t\) is in seconds. (a) What is the smallest positive value of \(x\) that corresponds to a node? Beginning at \(t=0\), what is the value of the (b) first, (c) second, and (d) third time the particle at \(x=0\) has zero velocity?

Step-by-Step Solution

Answer

(a) 0.5 m, (b) 0.25 s, (c) 0.5 s, (d) 0.75 s.

Step 1: Combine the two waves

The standing wave is formed by the superposition of the two traveling waves. We can express the combined wave as: \[ y = y_1 + y_2 = 0.050\cos(\pi x - 4\pi t) + 0.050\cos(\pi x + 4\pi t) \] Using the trigonometric identity: \[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \] we find: \[ y = 2 \times 0.050 \cos(\pi x) \cos(-4\pi t) = 0.10 \cos(\pi x) \cos(4\pi t) \] This is the equation of the standing wave.

Step 2: Identify node locations

Nodes in standing waves occur where the amplitude of the wave is zero. From the equation \[ y = 0.10 \cos(\pi x) \cos(4\pi t) \] this requires \( \cos(\pi x) = 0 \). \( \cos(\pi x) = 0 \) implies \( \pi x = (2n+1)\frac{\pi}{2} \) where \( n \) is an integer. Therefore, \[ x = (2n+1)\frac{1}{2} \] The smallest positive value corresponds to \( n = 0 \), yielding \( x = \frac{1}{2} \) m.

Step 3: Find timing of zero velocity at x=0

The particle at \( x = 0 \) will have a standing wave equation \( y = 0.10 \cos(4\pi t) \). The velocity \( v = \frac{dy}{dt} = -0.40\pi \sin(4\pi t) \). The velocity is zero when \( \sin(4\pi t) = 0 \), which gives: \[ 4\pi t = n\pi \] or \( t = \frac{n}{4} \) with \( n \) as an integer. For the first time (t > 0), \( n=1 \), \( t = \frac{1}{4} \) s.

Step 4: Determine second timing of zero velocity at x=0

Continuing the pattern \( t = \frac{n}{4} \), for the second time, choose \( n = 2 \). Thus, \[ t = \frac{2}{4} = \frac{1}{2} \] s.

Step 5: Determine third timing of zero velocity at x=0

Using the formula \( t = \frac{n}{4} \), for the third occurrence, set \( n = 3 \). Therefore, \[ t = \frac{3}{4} \] s.

Key Concepts

Superposition of Waves

The idea behind superposition is that when two or more waves overlap, their displacements at any point add together. When dealing with wave superposition, especially in the context of standing waves, it's important to understand how waves interact.

• In our exercise, two transverse traveling waves were combined: \[ y_1 = 0.050 \cos(\pi x - 4\pi t) \] and \[ y_2 = 0.050 \cos(\pi x + 4\pi t) \]. When these two waves move in opposite directions, they create a standing wave pattern through their superposition.
• This is achieved using the formula: \[ y = y_1 + y_2 = 2 \times 0.050 \cos(\pi x) \cos(4\pi t) \]. This equation represents the combined wave as a standing wave.

The fact that they move in opposite directions is key for creating nodes and antinodes in the standing wave pattern.

Node Calculation

In standing waves, nodes are points that remain at rest, meaning there is no movement at these locations. Nodes occur where the wave's amplitude is zero. To find these points, we analyze the standing wave equation: \[ y = 0.10 \cos(\pi x) \cos(4\pi t) \].

• A node exists when \( \cos(\pi x) = 0 \), meaning the wave's displacement is null. This condition is met when: \[ \pi x = (2n+1)\frac{\pi}{2} \] where \( n \) is an integer.
• Simplifying this gives us the node locations as: \[ x = (2n+1)\frac{1}{2} \], yielding the smallest positive value at \( x = \frac{1}{2} \) meters for \( n = 0 \).

This result is crucial for understanding how standing waves are structured, as nodes and antinodes define their patterns.

Wave Equation

A wave equation in general describes how a wave propagates through space and time. For this exercise, the equation of the standing wave derived from superposition reads: \[ y = 0.10 \cos(\pi x) \cos(4\pi t) \].

• This equation is particularly significant because it reveals two things: - The term \( \cos(\pi x) \) shows how the wave varies with position, highlighting where nodes and antinodes occur spatially. - The term \( \cos(4\pi t) \) sheds light on how the wave oscillates over time at each point in space.

Understanding this wave equation allows one to predict the motion and structure of a standing wave, giving insights into wave behavior at any given time and position.

Velocity of Wave Particles

The velocity of particles in standing waves is vital for grasping how energy is transferred over a medium, even if the position doesn't visibly change. In our example:

• The particle velocity for a point at \( x = 0 \) is defined by \( v = \frac{dy}{dt} = -0.40\pi \sin(4\pi t) \). This indicates that the velocity changes as a sine function of time.
• A particle's velocity is zero whenever: \( \sin(4\pi t) = 0 \), leading to: \[ 4\pi t = n\pi \] or \( t = \frac{n}{4} \) seconds, where \( n \) is an integer.

For instance, the first, second, and third times when velocity at \( x=0 \) is zero occur at \( t = \frac{1}{4}, \frac{1}{2}, \frac{3}{4} \) seconds respectively. This detail illustrates periodic behavior in a standing wave, emphasizing periods of rest and motion.