Question

Zwei als unendlich lang gedachte Seile werden am linken Ende mit der Amplitude A und der Frequenz \(\nu\) erregt. Geben Sie die Wellenfunktion an für Seil a) \(\quad A=0,5 \mathrm{~m} ; \quad \nu=5 \mathrm{~s}^{-1} ; \quad \lambda=1,2 \mathrm{~m}\) Seil b) \(\quad A=0,2 \mathrm{~m} ; \quad \nu=0,8 \mathrm{~s}^{-1} ; \quad \lambda=4,0 \mathrm{~m}\) Ist die Wellengeschwindigkeit für beide Seile gleich?

Short answer

The wave functions for rope a and b are \(y_a(x,t) = 0.5\sin\left(\frac{2\pi}{1.2}x - 2\pi\cdot5t\right)\) and \(y_b(x,t) = 0.2\sin\left(\frac{2\pi}{4.0}x - 2\pi\cdot0.8t\right)\) respectively. Also, the wave speeds of both ropes are not equal. For rope a it is 6 m/s and for rope b it is 3.2 m/s.

Step-by-step solution

Derive the Wave Function for Rope a

The wave function for a simple harmonic wave can be represented as \(y(x,t) = A\sin(kx - \omega t)\) where \(k = \frac{2\pi}{\lambda}\) is the wave number and \(\omega = 2\pi\nu\) is the angular frequency. Given the amplitude A = 0.5 m, frequency \(\nu = 5 s^{-1}\), and wavelength \(\lambda = 1.2 m\), substitute these values into the equation to derive the wave function for rope a: \(y_a(x,t) = 0.5\sin\left(\frac{2\pi}{1.2}x - 2\pi\cdot5t\right)\)

Derive the Wave Function for Rope b

Similarly, substitute the given amplitude A = 0.2 m, frequency \(\nu = 0.8 s^{-1}\), and wavelength \(\lambda = 4.0 m\) into the equation to derive the wave function for rope b: \(y_b(x,t) = 0.2\sin\left(\frac{2\pi}{4.0}x - 2\pi\cdot0.8t\right)\)

Calculate the Wave Speeds for Both Ropes

The wave speed (v) can be calculated using the formula \(v = \lambda\nu\). Hence, the wave speeds for rope a and b would be \(v_a = 1.2 \times 5 = 6 \text{ m/s}\) and \(v_b = 4.0 \times 0.8 = 3.2 \text{ m/s}\) respectively.

Compare the Wave Speeds

The wave speed for rope a is 6 m/s and for rope b is 3.2 m/s. Thus, the wave speeds for both ropes are not equal.

Key concepts

Harmonic Wave

In physics, a harmonic wave is characterized by a repetitive oscillation that is sinusoidal in time and space. This means that the shape of the wave if plotted, would look like the familiar crests and troughs of a sine wave. Harmonic waves are fundamental in understanding how energy travels through various mediums, such as the vibration in strings, sound in air, or light.

To describe a harmonic wave mathematically, one can use the wave function, often represented by the equation: \( y(x,t) = A\sin(kx - \omega t) \), where:

• \( y(x,t) \) is the displacement of the wave at position \( x \) and time \( t \).
• \( A \) is the amplitude, which measures the maximum displacement of the wave from its rest position.
• \( k \) is the wave number and relates to the wavelength \( \lambda \) as \( k = \frac{2\pi}{\lambda} \).
• \( \omega \) is the angular frequency, explaining how fast the wave oscillates and connects to the frequency \( u \) by \( \omega = 2\piu \).

The harmonicity ensures that the wave is periodic in space and time, making it predictable and easy to analyze. As seen in the exercise, harmonic waves can vary based on their amplitude and frequency, which directly influence the wave’s shape and energy.

Angular Frequency

In the realm of wave mechanics, angular frequency is a measure of how rapidly the wave oscillates in time. It's denoted by the symbol \( \omega \) and is related to the ordinary frequency \( u \) that we are more familiar with, such as in the number of vibrations per second. The relationship between them is given by the equation: \( \omega = 2\piu \).

Angular frequency is measured in radians per second, as it considers the angle travelled by a point on the wave in a unit of time. Diving into the exercise, we've seen how angular frequency is derived from the given frequency, with \( \omega = 2\pi\times5 \) for rope a and \( \omega = 2\pi\times0.8 \) for rope b. It represents the speed at which the wave's phase oscillates, which affects how the wave behaviour is described mathematically and perceived physically. This concept is vital when determining wave properties and how they evolve over time.

Wave Speed

The wave speed is a fundamental property of waves that represents how fast the wave travels through the medium. In our exercise, we calculate wave speed using the equation: \( v = \lambdau \), meaning the wave speed \( v \) equals the product of the wavelength \( \lambda \) and the frequency \( u \).

The term 'speed' here refers to the speed of the wavefront, which is the leading edge of one cycle of the wave, as it moves through space. For example, a wave with a long wavelength and high frequency will have a higher speed than one with a shorter wavelength and lower frequency, all else being equal. In the context of our exercise, the wave speeds for rope a and b are \( v_a = 6 \text{ m/s} \) and \( v_b = 3.2 \text{ m/s} \) respectively, showing that even though the two waves might have the same origin, their properties can result in different wave speeds. This concept is crucial in many areas of physics and engineering, as it influences how energy and information travel and how different media react to wave propagation.