Problem 41 - Master Resource Book in JEE Main Physics

Question

The equations of displacement of two waves are given as \(y_{1}=10 \sin (3 \pi t+\pi / 3)\) $$ y_{2}=5(\sin 3 \pi t+\sqrt{3} \cos 3 \pi t) $$ then what is the ratio of their amplitude? (a) \(1: 2\) (b) \(2: 1\) (c) \(1: 1\) (d) None of these

Step-by-Step Solution

Answer

(c) 1:1

Step 1: Identify Components of First Wave

The equation for the first wave is given by \( y_1 = 10 \sin (3 \pi t + \pi / 3) \). This is in the form of a simple sinusoidal wave with amplitude. The amplitude of the wave \( y_1 \) is the coefficient of the sine function, which is 10.

Step 2: Convert Second Equation to Sine-Cosine Form

For the second equation \( y_2 = 5(\sin 3\pi t + \sqrt{3} \cos 3\pi t) \), observe that it is a linear combination of \( \sin(x) \) and \( \cos(x) \). This can be rewritten in the form \( A \sin(x + \phi) \) where \( A \) is the amplitude.

Step 3: Calculate Amplitude of Second Wave

To find the amplitude of \( y_2 \), use the identity for rewriting as one sinusoidal wave: \( A = \sqrt{B^2 + C^2} \) where \( B = 1 \) and \( C = \sqrt{3} \). Thus, \( A = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2. \) However, \( y_2 \) includes a 5 outside, so the amplitude is \( 5 \times 2 = 10. \)

Step 4: Determine Ratio of Amplitude

The amplitude of both waves \( y_1 \) and \( y_2 \) is 10. Therefore, the ratio of their amplitudes is \( \frac{10}{10} = 1:1 \).

Key Concepts

Sinusoidal Waves

Sinusoidal waves are a fundamental concept in understanding wave mechanics and oscillations. They describe a type of continuous, smooth, repetitive oscillation resembling the sine or cosine function shape. This pattern is widely used to model various physical phenomena, including sound and light waves. The general form of a sinusoidal wave can be expressed as \( y = A \sin (kx + \omega t + \phi) \) or \( y = A \cos (kx + \omega t + \phi) \), where:

• \( A \) is the wave's amplitude, indicating the peak value of the wave.
• \( k \) is the wave number, relating to the wavelength.
• \( \omega \) is the angular frequency, dictating how fast the wave oscillates in time.
• \( \phi \) represents the phase shift, altering the wave's starting position.

Sinusoidal waves are significant because they provide a simple yet powerful way to describe complex periodic behaviors. They serve as building blocks for more complex waves due to the principle of superposition.

Amplitude Ratio

The amplitude ratio is a comparison between the amplitudes of two or more waves. For instance, in our problem, we are comparing two waves to understand their relative strength or intensity. The amplitude of a wave is the maximum displacement from its equilibrium position, reflecting how much energy the wave carries.Calculating the amplitude ratio involves determining the amplitudes of the waves involved. Once the amplitudes \( A_1 \) and \( A_2 \) are known, the ratio is simply \( \frac{A_1}{A_2} \). In the provided exercise, both waves \( y_1 \) and \( y_2 \) have an amplitude of 10. Therefore, their amplitude ratio is \( 1:1 \).
By examining amplitude ratios, we can determine whether waves will interfere constructively or destructively when they superpose. This concept is crucial in applications like sound engineering, where managing wave strength leads to desired audio effects.

Trigonometric Functions

Trigonometric functions are the backbone for describing periodic phenomena such as waves. The basic trigonometric functions—sine and cosine—are crucial in modeling sinusoidal waves.The sine function, \( \sin \), gives the y-coordinate of a point on the unit circle as it sweeps through an angle. Cosine, \( \cos \), similarly provides the x-coordinate. These functions repeat values every \( 2\pi \) radians, giving them their periodic nature.

• \( \sin(x) \) describes the oscillation starting from the origin, moving towards its maximum after a quarter period.
• \( \cos(x) \) starts at its maximum value and descends to the origin after a quarter period.

Given these properties, by expressing a wave equation in terms of sine and cosine functions, we can explore its behavior, such as amplitude, phase shift, and oscillation frequency. In our case, the wave equation \( y_2 \) is transformed into a single sinusoidal form to calculate its amplitude efficiently through the identity \( A = \sqrt{B^2 + C^2} \), where \( B \) and \( C \) relate to the sine and cosine coefficients, respectively.