Question

State (i) the amplitude and (ii) the angular frequency of the following waves: (a) \(y=2 \sin 5 t\) (b) \(y=3 \cos 6 t\) (c) \(y=\sin \frac{t}{2}\) (d) \(y=\cos \frac{4 t}{3}\) (e) \(y=\frac{3}{2} \sin \frac{2 t}{3}\)

Short answer

Question: Identify the amplitude and angular frequency of the following wave functions: (a) \(y=2 \sin 5 t\) (b) \(y=3 \cos 6 t\) (c) \(y=\sin \frac{t}{2}\) (d) \(y=\cos \frac{4 t}{3}\) (e) \(y=\frac{3}{2} \sin \frac{2 t}{3}\) Answer: (a) Amplitude: 2, Angular Frequency: 5 (b) Amplitude: 3, Angular Frequency: 6 (c) Amplitude: 1, Angular Frequency: 1/2 (d) Amplitude: 1, Angular Frequency: 4/3 (e) Amplitude: 3/2, Angular Frequency: 2/3

Step-by-step solution

(a) Identify Amplitude and Angular Frequency

For wave (a): \(y=2 \sin 5 t\) Here, the wave function is in the form \(y=A \sin(\omega t)\). Comparing it with the general equation, we obtain: Amplitude \(A = 2\) Angular Frequency \(\omega = 5\) So the amplitude is \(2\) and the angular frequency is \(5\).

(b) Identify Amplitude and Angular Frequency

For wave (b): \(y=3 \cos 6 t\) Here, the wave function is in the form \(y=A \cos(\omega t)\). Comparing it with the general equation, we obtain: Amplitude \(A = 3\) Angular Frequency \(\omega = 6\) So the amplitude is \(3\) and the angular frequency is \(6\).

(c) Identify Amplitude and Angular Frequency

For wave (c): \(y=\sin \frac{t}{2}\) Here, the wave function is in the form \(y= \sin(\omega t)\). Comparing it with the general equation, we obtain: Amplitude \(A = 1\) Angular Frequency \(\omega = \frac{1}{2}\) So the amplitude is \(1\) (since it's not given, we assume it to be 1) and the angular frequency is \(\frac{1}{2}\).

(d) Identify Amplitude and Angular Frequency

For wave (d): \(y=\cos \frac{4 t}{3}\) Here, the wave function is in the form \(y= \cos(\omega t)\). Comparing it with the general equation, we obtain: Amplitude \(A = 1\) Angular Frequency \(\omega = \frac{4}{3}\) So the amplitude is \(1\) (since it's not given, we assume it to be 1) and the angular frequency is \(\frac{4}{3}\).

(e) Identify Amplitude and Angular Frequency

For wave (e): \(y=\frac{3}{2} \sin \frac{2 t}{3}\) Here, the wave function is in the form \(y=A \sin(\omega t)\). Comparing it with the general equation, we obtain: Amplitude \(A = \frac{3}{2}\) Angular Frequency \(\omega = \frac{2}{3}\) So the amplitude is \(\frac{3}{2}\) and the angular frequency is \(\frac{2}{3}\).

Key concepts

Amplitude

Amplitude refers to the maximum displacement from the rest position in a wave. It's the 'height' of the wave and is a measure of the wave's intensity or strength. In our exercise, we identified the amplitude by comparing the wave function to the general form of a sine or cosine wave. For example, in the wave equation \(y=2 \sin 5 t\), the amplitude is the coefficient before the sine function, which is 2 in this case.

It's important to note that a larger amplitude means that the wave carries more energy. Imagine throwing a pebble into a pond; the larger the pebble, the larger the ripples, which represent the amplitude of the water waves. In sound waves, amplitude is related to the volume of the sound; louder sounds have greater amplitudes. Similarly, in light waves, the amplitude is associated with the brightness of the light.

Sine Wave

A sine wave is a smooth, repetitive oscillation that is mathematically described by the sine function. It is characterized by its sinusoidal shape and is one of the most fundamental waveforms in physics and mathematics. In the context of our exercise, the sine wave equation takes the form \(y=A \sin(\omega t)\), where \(A\) is the amplitude and \(\omega\) is the angular frequency. The variable \(t\) typically represents time.

The sine wave is significant in many fields, including acoustics, electronics, and signal processing. When you listen to a clear, pure note played on a tuning fork or a whistle, you're hearing a sound that closely resembles a sine wave. The elegance of sine waves lies not only in their simplicity but also in their function as building blocks; complex waves can often be constructed by adding multiple sine waves of differing amplitudes and frequencies.

Cosine Wave

Cosine waves are similar to sine waves with the key difference being a phase shift. While a sine wave starts from zero, a cosine wave starts from its maximum amplitude value. In our exercise, we were given examples in the form \(y=A \cos(\omega t)\), indicating a cosine wave where \(A\) represents the amplitude and \(\omega\) is the angular frequency.

The cosine function is often used in trigonometry and physics to model wave behavior, just like the sine function. You can think of a cosine wave as a sine wave that has been shifted in the horizontal direction. Despite this phase shift, both types of waves have similar properties and mathematical significance. They are also both periodic, meaning they repeat their values in regular intervals over time. In practical applications, such as alternating current (AC) electricity, cosine waves can be used to describe voltage or current variations over time.