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Understanding how to calculate the wavelength of a wave is crucial in wave mechanics. The wavelength (\( \lambda \)) represents the distance between two consecutive points that are in phase, such as the distance between two peaks or troughs of a wave.
In our problem, we have a wave described by the equation:\[D = 0.22 \sin(5.6x + 34t)\]
The wave number (\( k \)) is given as 5.6. We can find the wavelength using the formula:\[\lambda = \frac{2\pi}{k}\]
Substituting\( k = 5.6 \), we get:\[\lambda = \frac{2\pi}{5.6} \approx 1.12 \text{ meters}\]
Thus, the wavelength of this wave is approximately 1.12 meters.

Frequency refers to how many cycles or oscillations a wave completes in one second. It is measured in Hertz (Hz). Knowing the frequency helps us understand how often the wave peaks.
In the given equation:\[D = 0.22 \sin(5.6x + 34t)\]
The angular frequency (\( \omega \)) is provided as 34. We can calculate the frequency (\( f \)) using the relationship:\[\omega = 2\pi f\]
Rearranging for\( f \), gives:\[f = \frac{\omega}{2\pi} = \frac{34}{2\pi} \approx 5.41 \text{ Hz}\]
This tells us that the wave completes approximately 5.41 cycles per second.

Wave velocity is the speed and direction in which the wave travels through the medium. It can be determined from the frequency and wavelength.
The wave velocity (\( v \)) is calculated using the equation:\[v = f\lambda\]
With\( f = 5.41 \text{ Hz} \) and\( \lambda = 1.12 \text{ meters} \), we find:\[v = 5.41 \times 1.12 \approx 6.06 \text{ m/s}\]
This means the wave travels at a speed of approximately 6.06 meters per second. The direction of travel is determined from the equation whether the terms involving\( x \)t and\( t \) are positive or negative. In this case, since both\( 5.6x \) and\( 34t \) are positive, the wave moves in the negative x-direction.

Amplitude is the maximum displacement of the particles from their equilibrium position in the medium through which the wave travels. It gives us an idea of the wave's energy.
In our wave equation:\[D = 0.22 \sin(5.6x + 34t)\]
The amplitude (\( A \)) is represented by the coefficient before the sine function, which is 0.22.
This means the particles move 0.22 meters from their rest position in both directions.

• The larger the amplitude, the more energy the wave carries.
• In practical terms, amplitude affects the loudness of sound waves and brightness of light waves.

Particles in the medium through which the wave travels oscillate back and forth. Their speed varies over time during this motion.
In the context of our wave, the particle speed (\( v_{\text{particle}} \)) can be calculated by differentiating the displacement with respect to time:\[v_{\text{particle}} = A \omega \cos(kx + \omega t)\]
Given\( A = 0.22 \) and\( \omega = 34 \), the maximum particle speed occurs when\( \cos \) is 1, so:\[v_{\text{particle, max}} = 0.22 \times 34 = 7.48 \text{ m/s}\]
This maximum value indicates how fast particles can move in the medium.
The minimum speed is 0 m/s and occurs when the particle reaches its equilibrium position, where there is no displacement.