The Equation of a Wave and Its Parameters

In the realm of wave analysis, the given equation y(x, t) 5.5 sin(pi cdot 0.020 cdot x - 4.0t) provides a detailed description of a wave's behavior over time and space. This article delves into the intrinsic elements that characterize this wave: amplitude, wavelength, frequency, and wave speed. By understanding these parameters, we can better comprehend the wave's dynamics.

Understanding the Wave Equation

The equation for the wave in question is y(x, t) 5.5 sin(pi cdot 0.020 cdot x - 4.0t), where y and x are in centimeters (cm) and t is in seconds (s). Let's explore the wave parameters in detail.

Amplitude

The amplitude A is the coefficient in front of the sine function, which describes the maximum displacement of the wave from its equilibrium position. From the given equation, we can determine the amplitude as follows:

A  5.5 , text{cm}

This means that the wave oscillates between -5.5 cm and 5.5 cm from its equilibrium position.

Wavelength

The wavelength lambda is the distance between two consecutive identical points on a wave. It is related to the wave number k by the equation lambda frac{2pi}{k}.

Using the wave equation, we can identify the wave number k as:

k  pi cdot 0.020 , text{m}^{-1}  0.020pi , text{cm}^{-1}

Substituting k into the equation for wavelength:

lambda  frac{2pi}{0.020pi}  frac{2}{0.020}  100 , text{cm}

Therefore, the wavelength of the wave is 100 cm.

Frequency

The frequency f is the number of wave cycles that pass a fixed point per second. It is related to the angular frequency omega by the equation f frac{omega}{2pi}.

From the given equation, the angular frequency omega is:

omega  4.0 , text{s}^{-1}

Calculating the frequency:

f  frac{4.0}{2pi} approx frac{4.0}{6.2832} approx 0.637 , text{Hz}

The frequency of the wave is approximately 0.637 Hz.

Wave Speed

The wave speed v is the distance traveled by a wave per unit of time. It can be calculated using the relationship between frequency and wavelength:

v  f lambda

Substituting the values of f and lambda:

v  0.637 , text{Hz} times 100 , text{cm} approx 63.7 , text{cm/s}

Hence, the wave speed is approximately 63.7 cm/s.

Summary of Parameters

Amplitude: 5.5 cm
This is the maximum value attained by y. Wavelength: 100 cm
This is the distance between two consecutive identical points on the wave. Frequency: Approximately 0.637 Hz
This is the number of wave cycles per second. Wave Speed: Approximately 63.7 cm/s
This is the distance traveled by the wave per unit of time.

Additional Insights on Wave Parameters

The amplitude of the wave is 5.5 cm, indicating the maximum displacement from the equilibrium position. The wavelength is 100 cm, which means that the wave repeats its pattern every 100 cm along the x-axis. The frequency is about 0.637 Hz, which tells us that the wave completes approximately 0.637 cycles per second. The wave speed of 63.7 cm/s indicates how fast the wave travels along the x-axis.

Using the principles described above, we can understand and analyze many other wave equations by identifying the key parameters such as amplitude, wavelength, frequency, and wave speed. These parameters provide a comprehensive picture of the wave's behavior, allowing for a deeper understanding of the physical phenomena at play.

Conclusion

In summary, the given wave equation provides us with the necessary parameters to fully understand the behavior of the wave. By calculating the amplitude, wavelength, frequency, and wave speed, we can describe the wave completely and accurately. This understanding is crucial for applications in physics, engineering, and other fields where wave analysis is essential.