Question

A wave \(\mathrm{y}=\mathrm{a} \sin (\omega \mathrm{t}-\mathrm{kx})\) on a string meets with another wave producing a node at \(\mathrm{x}=0 .\) Then the equation of the unknown wave is \(\ldots \ldots \ldots\) (A) \(y=a \sin (\omega t+k x)\) (B) \(\mathrm{y}=-\mathrm{a} \sin (\omega \mathrm{t}+\mathrm{kx})\) (C) \(\mathrm{y}=\mathrm{a} \sin (\omega \mathrm{t}-\mathrm{kx})\) (D) \(\mathrm{y}=-\mathrm{a} \sin (\omega \mathrm{t}-\mathrm{kx})\)

Short answer

The equation of the unknown wave is \(y = -a \sin(\omega t - kx)\) (option D).

Step-by-step solution

Superposition of waves

When two waves interfere, their amplitudes add together by the principle of superposition. Let the unknown wave be Y(x, t), then the resulting wave after the interference will be y(x, t) + Y(x, t). #Step 2: Apply condition of node at x = 0#

Node condition

If there is a node at x = 0, then the total displacement at this point must be zero. We can write this as y(0, t) + Y(0, t) = 0. #Step 3: Substitute x = 0 into the condition and solve for Y(0, t)#

Substitute and solve

Using the equation of the first wave, we can write y(0, t) = a sin(ωt - k * 0) = a sin(ωt). Substitute this into the node condition: a sin(ωt) + Y(0, t) = 0 ⇒ Y(0, t) = -a sin(ωt) #Step 4: Identify the matching wave equation#

Compare with the given options

Now we compare our result, Y(0, t) = -a sin(ωt), with the given options to find the most suitable option. (A) y=a sin(ωt + kx): The sine term is not inverted, so this is not the correct option. (B) y=-a sin(ωt + kx): The sine term is inverted, but the wave is traveling in the same direction as the first wave, so this is not the correct option. (C) y=a sin(ωt - kx): The equation is the same as the first wave, so this is not the correct option. (D) y=-a sin(ωt - kx): The sine term is inverted, and the wave travels in the same direction as the first wave, so this is the correct option. Thus, the equation of the unknown wave is \(y = -a \sin(\omega t - kx)\) (option D).

Key concepts

Wave Interference

Wave interference occurs when two or more waves meet while traveling along the same medium. Different types of interference can happen based on the phase relationship of the waves:

• Constructive Interference: This occurs when waves meet in phase, meaning their crests and troughs align. They add up to produce a wave of greater amplitude.
• Destructive Interference: This happens when waves meet out of phase, meaning the crest of one wave aligns with the trough of another. The resulting amplitude is reduced, and can even become zero, leading to a point known as a node.

In our exercise, by using the principle of superposition, the superimposition of two waves resulted in an interference pattern featuring nodes, one of which occurred at the point where the total displacement was zero.

Nodes in Wave Motion

Nodes are points along a medium where no movement occurs due to destructive interference. They are crucial in the formation of standing waves, where certain points remain stationary, effectively splitting the wave into segments of motion.

• For a string fixed at both ends, nodes are equally spaced.
• At a node, the displacement due to individual waves exactly cancels out, resulting in zero net displacement.

In the exercise, placing a node at \(x = 0\) implies that the sum of the waves' amplitudes at this point is zero. This means one wave must have an inverted displacement compared to the other, hence leading us to find the proper equation that represents the unknown wave causing this node.

Wave Equation

A wave equation is a mathematical equation that describes the progression of waves through a medium. The wave equation of a single wave is usually represented as:\[ y(x,t) = a \sin(\omega t - kx) \]Here, \(a\) is the amplitude, \(\omega\) is the angular frequency, and \(k\) is the wave number. The negative sign in the argument \(\omega t - kx\) indicates a wave moving in the positive x direction.In the case of the exercise provided, solving the node condition requires us to explore how a wave can lead to a node when colliding with another. By considering the principle of superposition, the interfering wave must have the form:\[ Y(0, t) = -a \sin(\omega t) \]The negatives in this formula indicate that the interfering wave is exactly out of phase with the first, yet traveling in the same direction, aligning it with the characteristics provided by option \(D\). This illustrates that the phase opposition leads to destructive interference needed to form nodes in wave motion.