Engineering physics - Waves Video 3

Summary

In this lecture, the concepts of wavelength and time period in wave motion are explained. The wave traveling along the positive x-direction is represented by a sine function, with the amplitude and propagation constants discussed. The wavelength is defined as the distance after which the wave function repeats itself, equal to 2π/k. The time period, which is the time after which the wave function repeats, is derived as 2π/ω, linking angular frequency with the wave's velocity. The relationship between frequency, velocity, and wavelength is established, concluding with the equation v = νλ for the velocity of a wave.

Takeaways

• 😀 A wave traveling in the positive x-direction can be represented by the wave function: ψ(x,t) = A sin(kx - vt), where A is the amplitude, k is the wave number, and v is the wave velocity.
• 😀 The position of the wave at a given point can be analyzed by considering the displacement of the wave at different values of x and t.
• 😀 When the wave function is evaluated at a position shifted by 2π/k, the wave function repeats, showing that the wave has a spatial periodicity called the wavelength, λ = 2π/k.
• 😀 Wavelength (λ) represents the distance over which the wave function repeats itself in space, and it is defined as λ = 2π/k.
• 😀 Time period (T) refers to the time after which the wave function repeats at a given position, and it can be defined as T = 2π/ω, where ω is the angular frequency.
• 😀 The angular frequency ω is related to the wave velocity v and the wave number k by the equation ω = k * v.
• 😀 The time period can also be expressed as T = 2π / (k * v), linking it to the wave velocity and wave number.
• 😀 The relationship between frequency (ν) and time period is given by T = 1/ν, meaning that time period is the reciprocal of frequency.
• 😀 The velocity of the wave (v) is the product of the frequency (ν) and wavelength (λ), which is represented as v = ν * λ.
• 😀 The key takeaway is that the velocity of a wave can be expressed as the product of its frequency and wavelength, which is a fundamental concept in wave theory.

Q & A

What does the wave function for a wave traveling along the positive x direction look like?

The wave function for a wave traveling along the positive x direction can be represented as psi(x, t) = A * sin(kx - vt), where A is the amplitude of the wave, k is the wave number, x is the position, v is the velocity, and t is the time.

What does the variable 'A' in the wave function represent?

The variable 'A' in the wave function represents the amplitude of the wave, which defines the maximum displacement of points on the wave from their equilibrium position.

How is the wave function affected when x is increased by 2π/k?

When x is increased by 2π/k, the wave function repeats itself, meaning the displacement at the new position is the same as at the original position. Mathematically, psi(x + 2π/k, t) equals psi(x, t).

What is the wavelength of a wave and how is it related to the wave function?

The wavelength (λ) of a wave is the distance after which the wave function repeats itself. It is given by λ = 2π/k, where k is the wave number. The wave function repeats in space after a distance of λ.

How is the time period of a wave defined?

The time period (T) of a wave is the time after which the wave function repeats itself. Mathematically, it is given by T = 2π/ω, where ω is the angular frequency of the wave.

How is the angular frequency (ω) related to the wave number (k) and the velocity (v) of the wave?

The angular frequency ω is related to the wave number k and the velocity v by the equation ω = k * v.

What is the relationship between the time period and frequency of a wave?

The time period (T) and frequency (ν) of a wave are inversely related. The time period is the reciprocal of the frequency, i.e., T = 1/ν.

What does the equation v = ν * λ represent?

The equation v = ν * λ represents the velocity (v) of a wave, where ν is the frequency and λ is the wavelength. It shows that the velocity of a wave is the product of its frequency and wavelength.

What does the propagation constant (k) represent in wave motion?

The propagation constant (k) is a measure of the wave number, representing the spatial frequency of the wave. It is given by k = 2π/λ, where λ is the wavelength.

How is the wave function psi(x, t) affected by changes in time?

When time is increased by 2π/ω, the wave function repeats itself, meaning psi(x, t + 2π/ω) equals psi(x, t). This shows the periodic nature of the wave with respect to time.