Question

Show by examining the position of the nodes that \(\operatorname{Re}\left[A_{+} e^{i(k x-\omega t)}\right]\) and \(\operatorname{Re}\left[A_{-} e^{i(-k x-\omega t)}\right]\) represent plane waves moving in the positive and negative \(x\) directions, respectively. The notation Re[ ] refers to the real part of the function in the brackets.

Short answer

In conclusion, by examining the position of the nodes, we can show that \(\operatorname{Re}\left[A_{+} e^{i(k x-\omega t)}\right]\) represents a plane wave moving in the positive x direction, as its expression for \(x\) increases with time. On the other hand, \(\operatorname{Re}\left[A_{-} e^{i(-k x-\omega t)}\right]\) represents a plane wave moving in the negative x direction, as its expression for \(x\) decreases with time.

Step-by-step solution

Identify the Real Part of the Complex Plane Waves

To find the real part of the complex plane waves, we can use Euler's formula: \(e^{ix} = \cos{x} + i \sin{x}\). Applying this formula, we get: 1. \(\operatorname{Re}\left[A_{+} (\cos{(kx - \omega t)} + i\sin{(kx - \omega t)})\right] = A_{+}\cos{(kx - \omega t)}\) 2. \(\operatorname{Re}\left[A_{-} (\cos{(-kx - \omega t)} + i\sin{(-kx - \omega t)})\right] = A_{-}\cos{(-kx - \omega t)}\)

Find the Nodes

Now we will find the nodes of each plane wave - the points where the amplitude of the wave is zero. To find these points, we set the real part of the complex plane wave to zero: 1. \(A_{+}\cos{(kx - \omega t)} = 0\) 2. \(A_{-}\cos{(-kx - \omega t)} = 0\) Now, let's solve both equations for \(x\): 1. \(kx - \omega t = \arccos{\left(\frac{0}{A_{+}}\right)} + 2n\pi\), where \(n\) is an integer. 2. \(-kx - \omega t = \arccos{\left(\frac{0}{A_{-}}\right)} + 2m\pi\), where \(m\) is an integer. Now, we can find x in terms of t for both equations: 1. \(x = \frac{1}{k}(\omega t + 2n\pi)\) 2. \(x = -\frac{1}{k}(\omega t + 2m\pi)\)

Analyze the Direction

Looking at the expressions for \(x\) in terms of \(t\), we can now determine the direction of the plane waves. 1. As \(t\) increases, \(x\) also increases because both \(k\) and \(\omega\) are positive. Hence, this wave moves in the positive x direction. 2. In contrast, as \(t\) increases, \(x\) decreases, since the expression for \(x\) has a negative sign in front of the term with \(t\). Hence, this wave moves in the negative x direction. In conclusion, \(\operatorname{Re}\left[A_{+} e^{i(k x-\omega t)}\right]\) represents a plane wave moving in the positive x direction, and \(\operatorname{Re}\left[A_{-} e^{i(-k x-\omega t)}\right]\) represents a plane wave moving in the negative x direction, as demonstrated by examining the position of their nodes.

Key concepts

Euler's Formula

Euler's formula is a fundamental principle in mathematics connecting complex exponentials with trigonometric functions. It states that for any real number \(x\), the formula \(e^{ix} = \cos{x} + i \sin{x}\) holds true. This equation allows us to express complex exponential functions in terms of sine and cosine, which are more familiar and easier to interpret.

In the context of physics, particularly in wave mechanics, Euler's formula helps in converting complex expressions into real components. When dealing with complex plane waves, like \(e^{i(kx - \omega t)}\), Euler's formula assists in identifying the real part of the wave, which is crucial for understanding the wave's physical manifestations. The real parts of these complex wave functions are often what we can observe or measure in experiments, making Euler's formula a bridge between abstract mathematics and real-world applications.

Complex Plane Waves

Complex plane waves are an important concept in wave physics. They are represented as exponential functions in the form \(A e^{i(kx - \omega t)}\) where \(A\) is the amplitude, \(k\) is the wave number, \(\omega\) is the angular frequency, \(x\) is the position, and \(t\) is time.

These expressions leverage complex numbers to simplify the mathematics involved in wave analysis. The imaginary unit \(i\) allows the phase of the wave to be represented as a sum of sine and cosine terms through Euler’s formula. Understanding complex plane waves involves extracting real-world predictions from their mathematical expressions.

• The real part of a complex wave (using Euler’s formula) gives insight into actual wave motion and characteristics.
• Complex plane waves make solving differentiable equations more manageable due to their compact notation.

Real Part of a Function

When analyzing complex functions, the real part represents the component free of imaginary numbers. In the case of a complex plane wave, using Euler's formula, we express the wave as \(A e^{i(kx - \omega t)} = A(\cos(kx - \omega t) + i \sin(kx - \omega t))\).

Extracting the real part involves focusing on the cosine component: \(A \cos(kx - \omega t)\). This part directly translates to the measurable property of the wave.

• The real part determines the observable variations in wave properties, like displacement or pressure in sound waves.
• Understanding the real part helps visualize how the wave propagates through space and time.

Making sense of the real part is crucial for correctly interpreting experimental data in many fields such as acoustics, optics, and engineering.

Wave Direction

The wave direction describes how a wave propagates through space. In the exercise, the direction of plane waves is deduced from the phase of the complex exponential function.

Different expressions of \(e^{i(kx - \omega t)}\) and \(e^{i(-kx - \omega t)}\) dictate the wave's movement direction:

• For \(e^{i(kx - \omega t)}\), as time \(t\) progresses, \(x\) increases, signifying the wave moves in the positive \(x\) direction.
• For \(e^{i(-kx - \omega t)}\), as \(t\) progresses, \(x\) decreases, which means the wave moves in the negative \(x\) direction.

Understanding wave direction is essential for predicting how waves send energy or information from one point to another in any medium.

Nodes in Wave Motion

Nodes in wave motion are points where the wave's amplitude is consistently zero. These points signify the positions of no movement in the wave, creating distinctive patterns in standing waves.

To find nodes in the plane waves described by \(\operatorname{Re}[A_{+}e^{i(kx - \omega t)}]\) and \(\operatorname{Re}[A_{-}e^{i(-kx - \omega t)}]\), we set the cosine expressions to zero:

• \(A_+ \cos(kx - \omega t) = 0\)
• \(A_- \cos(-kx - \omega t) = 0\)

This condition yields the positions of nodes at specific \(x\) values over time. Understanding these positions is critical in applications like acoustics and resonance, where node patterns determine the stability and interference of waves. Nodes can also assist in visualizing energy distribution along a medium.