Question

When two sound waves travel in the same direction in a medium, the displacement of a particle located at \(\mathrm{x}\) at time \(\mathrm{t}\) is given by \(\mathrm{y}_{1}=0.05 \cos (0.50 \mathrm{px}-100 \mathrm{pt}) \&\) \(\mathrm{y}_{2}=0.05 \cos (0.46 \mathrm{px}-92 \mathrm{pt})\), where \(\mathrm{y}_{1}, \mathrm{y}_{2}\) and \(\mathrm{x}\) are in meter and \(\mathrm{t}\) is in seconds. At \(\mathrm{x}=0\), how many times between \(\mathrm{t}=0\) and \(\mathrm{t}=1 \mathrm{~s}\) does the resultant displacement become zero ? (A) 46 (B) 50 (C) 92 (D) 100

Short answer

The number of times the resultant displacement becomes zero between \(t = 0\) and \(t = 1\) second is 2, which is not among the given options. There might be a mistake in the exercise or the provided answer choices.

Step-by-step solution

Calculate the resultant displacement

To find the resultant displacement, we will add the two given individual displacements y1 and y2, when x = 0: Resultant_displacement = y1 + y2 Substitute the given functions for y1 and y2, and the value of x = 0: Resultant_displacement = \(0.05 \cos(-100t) + 0.05 \cos(-92t)\)

Analyze the time gaps between zero crossings of the resultant displacement

We need to find how many times the resultant displacement becomes zero between t = 0s and t = 1s. To do this, we will first investigate the time gaps between two consecutive zero crossings of cosine functions. As we know from trigonometry that the cosine function crosses zero at \(\left(n + \frac{1}{2}\right) \pi\), where n = 0, 1, 2, ... Let m be the difference between the two consecutive zero crossings of the cosine functions: m = \(\left(n + \frac{1}{2}\right) \pi\) Solving for n: n = \(\frac{2m}{\pi}\) - \(\frac{1}{2}\) The difference between two consecutive zero crossings of y1 is \(-\frac{100\pi}{π}\) - \(\frac{100\pi}{3\pi}\) = 100 - 100/3 = 100/3 Similarily, the difference between two consecutive zero crossings of y2 is \(-\frac{92\pi}{π}\) - \(\frac{92\pi}{3\pi}\) = 92 - 92/3 = 92/3

Calculate the number of zero crossings in the given time interval

Now that we have the difference between consecutive zero crossings (100/3 for y1 and 92/3 for y2), we can find the number of times the resultant displacement becomes zero between t = 0s and t = 1s. For y1, it will be the number of integer values of n such that 100/3 * n lies between 0 and 1: There is 1 such value when n = 1 (100/3 * 1 = 100/3, which lies between 0 and 1) For y2, it will be the number of integer values of n such that 92/3 * n lies between 0 and 1: There is 1 such value when n = 1 (92/3 * 1 = 92/3, which lies between 0 and 1)

Determine the number of times the resultant displacement is zero

Since both y1 and y2 have only one-zero crossing within the time interval of 0 to 1 second, the total number of times the resultant_displacement becomes zero is: Number_of_zero_crossings = 1 (for y1) + 1 (for y2) Number_of_zero_crossings = 2 The right answer is not among the given options (A, B, C, D). Therefore, it's possible that there is a mistake in the exercise or in the answers provided.

Key concepts

Sound Waves

Sound waves are a type of mechanical wave that travels through a medium such as air, water, or solid objects. They are produced by vibrating objects, which cause the surrounding medium to vibrate as well. This vibration travels through the medium in waves. These waves are called longitudinal waves because the particles of the medium move parallel to the wave direction.

Some characteristics of sound waves include:

• Frequency: The number of wave cycles that pass a point in one second, measured in Hertz (Hz).
• Amplitude: The height of the wave, which determines the loudness of the sound.
• Wavelength: The distance between consecutive crests or troughs of a wave.
• Speed: The speed at which the wave travels through the medium, which depends on the properties of the medium.

In the given problem, sound waves travel in a medium, causing displacement in surroundings. The mathematical description uses trigonometric functions to represent these waves' behavior over time.

Wave Superposition

Wave superposition is a fundamental concept in wave behavior, describing how multiple waves interact with each other. When two or more waves meet, they overlap and combine to form a new wave pattern. This is known as superposition. The principle of superposition states that the resultant displacement at any point is the sum of the displacements due to each wave.

In the context of the exercise, the sound waves are represented by \( y_1 \) and \( y_2 \), and we are interested in their combined effect at a particular point. At \( x = 0 \), the problem involves adding these two wave functions to find the resultant wave. The equation becomes the sum \( y_1 + y_2 \). This combined wave can lead to constructive or destructive interference:

• Constructive interference occurs when the waves are in phase, and their amplitudes add up, producing a larger wave.
• Destructive interference happens when the waves are out of phase, canceling each other out to some degree, potentially leading to zero displacement.

Understanding superposition helps explain why waves can create complex patterns and behaviors when interacting.

Zero Crossings

Zero crossings refer to the points in time where a wave's displacement crosses the axis, moving from positive to negative or vice versa. These crossings are significant because they indicate where the wave or resultant wave has a displacement of zero.

In the exercise, finding zero crossings for the resultant wave function is crucial. Each crossing represents a moment where the displacement of the medium is exactly zero, meaning the waves may be canceling each other out completely at these times. Understanding zero crossings can provide insight into the wave's behavior without needing a complete picture of its full waveform.

To determine when these zero crossings occur, one can use the mathematical properties of trigonometric functions. Specifically, cosine functions, which are used to describe the wave displacements, can be analyzed at these points to solve for time intervals between crossings. This process involves calculating when the sum of the angles inside the trigonometric functions (cosines) equals a value that makes the resultant wave's amplitude zero, such as \( n\pi \), where \( n \) is an integer.

Trigonometric Functions

Trigonometric functions are mathematical functions related to the angles and sides of a triangle. They are fundamental in describing periodic phenomena, including waveforms such as sound waves. The most commonly used trigonometric functions are sine, cosine, and tangent. In wave problems, cosine and sine functions are primarily used due to their periodic nature.

In the given exercise, the displacement of the particles due to sound waves is represented using the cosine function, \( y_1 = 0.05 \cos(0.50 \pi x - 100 \pi t) \) and \( y_2 = 0.05 \cos(0.46 \pi x - 92 \pi t) \). These functions model how the wave moves and varies over time and space.

• Phase: The term inside the cosine represents the phase of the wave, influencing the wave's position and shape over time.
• Amplitude: The coefficient \( 0.05 \) in front of the cosine dictates the wave's maximum displacement, affecting the wave's strength.

Using trigonometric functions like cosine allows us to precisely calculate moments such as zero crossings, which play a role in understanding interaction points and interference patterns. Mastery of these functions is crucial for solving complex wave interference problems in physics.