Question

Pruve that fur any sine function \(\sin (k x+\phi)\) of wavelength shonter than \(2 a\), where \(a\) is the atomic spacing. there is a sine function with a wavelength longer than \(2 a\) that has the same values at the points \(x=a .20,3 a\). and so on. Wotr: It is probably easier to work with wave number than with wavelength. We seck to show that for every wave number greater than \(\pi / a\) there is an equivalent une less than \(\pi / a\).)

Short answer

The given sine function of wavelength shorter than \(2a\), can also be represented with an equivalent sine function of wavelength longer than \(2a\). The function has the same values at given points \(x=a, 2a, 3a, \...) because of the periodic property of the sine function. This was achieved by subtracting multiples of \(2\pi\) from the wave number, resulting in an equivalent wave number that fits within the required constraints.

Step-by-step solution

Understanding Sine function

For a sine function in the form \(\sin(kx + \phi)\), it refers to a sine wave which can be adjusted by the parameter \(k\) (wavenumber or angular frequency) and the phase shift \(\phi\). When it comes to a wave number \(k = 2\pi / \lambda\), where \(\lambda\) is wavelength of the sine wave. So, when \(k > \pi / a\) that implies \(\lambda < 2a\). The task is to find a equivalent sine function with \(k' < \pi / a\), which implies \(\lambda' > 2a\).

Establishing Identity

In order to find the equivalent wave number \(k'\), it's important to note that sine is a periodic function with a period of \(2\pi\). As sine values repeat every \(2\pi\), we can use this property to subtract a multiple of \(2\pi\) from the wave number \(kx\). This leaves us with the wave number \(k'x = kx - 2n\pi\), where \(n\) is any integer such that the \(k'x\) remains in the period of sine, ideally less than \(\pi / a\). Select \(n\) such that \(k' < \pi / a\), which would be equivalent to \(n > ka / 2 - 1 / 2\). Choose the smallest integer \(n\) that is greater than \(ka / 2 - 1 / 2\). Then \(k' = k - 2n / a\).

Demonstration of Identity

It can be demonstrated that the points \(x=a, 2a, 3a, \dots\) are the points on the curve where \(k' = k - 2n / a\) and \(kx + \phi = k'x + \phi\). This is because the integer \(n\) fits exactly into \(x/a\), leaving a remainder of zero. So, \((k - k')x = 2n\pi\), where the \(n\) is an integer, equals to \(2\pi\) times an integer, will result in zero, because of periodic property.So, there is indeed another wave with \(k' < \pi / a\) that takes on the same values at the given series of points \(x = a, 2a, 3a,...\) as the wave with \(k > \pi / a\).

Key concepts

Wavelength and Wave Number

Understanding the relationship between wavelength and wave number is crucial when studying wave phenomena in physics. The wavelength, denoted as \(\lambda\), is the distance over which the wave's shape repeats itself. In contrast, the wave number \(k\) is a measure of the number of wave cycles per unit distance, which can be expressed as \(k = \frac{2\pi}{\lambda}\).

• Higher wave numbers indicate shorter wavelengths and vice versa.
• The wave number is directly proportional to the frequency, with higher frequencies corresponding to larger wave numbers.

Considering the periodic nature of the sine function, an increased wave number translates to a higher frequency of oscillation in a given space and a decreased wavelength. Manipulating the wave number allows us to create waves that fit into specific spatial constraints, such as atomic spacings in a material.

In the context of the exercise, we are interested in finding pairs of sine functions where one function has a wavelength shorter than \(2a\) while its counterpart has a wavelength longer than \(2a\), yet both share the same values at specific points. This illustrates the versatility and duality present within wave properties in physics.

Periodic Functions

Sine functions are a prime example of periodic functions, which are functions that repeat their values in regular intervals or periods. The period of the sine function is \(2\pi\), meaning that the function will complete one full cycle and return to its initial value after its variable increases by \(2\pi\). This property is central to solving many problems in physics where periodicity plays a role.

• A sine wave completes a cycle over an interval of \(2\pi\).
• The periodic nature allows for the prediction of values at various points once a single period is known.

Referring to the given exercise, we take advantage of the sine function's periodicity to find another wave number that results in the same values at multiples of the atomic spacing \(a\). This hinges on the fact that any integer multiple of \(2\pi\) added or subtracted from the sine function's argument will not change its value, thereby enabling us to 'fold' one wave into another with a different wavelength but identical values at discrete points.

Sine Wave Properties

A sine wave, one of the fundamental shapes found in physics, displays several noteworthy properties. The amplitude represents the wave's peak value, the wavelength \(\lambda\) determines the distance between two adjacent peaks, and the frequency tells us how often the peaks pass a point over a given time. Additionally, the phase shift \(\phi\) allows the wave to be moved horizontally along the axis, altering where it begins within its cycle.

Characterizing Sine Waves

• The amplitude modulates the wave's height.
• The phase shift changes the starting position of the cycle.
• Frequency and wave number relate to how tightly the wave cycles are packed together.

In our exercise, the focus is on manipulating the wave number to adjust the wavelength and thus influence where the peaks occur relative to the atomic spacing \(a\). This manipulation demonstrates how sine waves can be modeled to fit very specific criteria, a concept often applied in signal processing, acoustics, and other physics applications.

Angular Frequency

The term angular frequency, usually denoted by \(\omega\), is used in physics to describe the rate of change of the phase of a sinusoidal waveform, or in other terms, how rapidly the argument of the sine or cosine functions increases over time. It is related to the wave number and frequency by the equation \(\omega = 2\pi f = kv\), where \(f\) is the linear frequency, and \(v\) is the wave's speed.

• Angular frequency is measured in radians per second.
• It dictates how quickly the wave oscillates in time.

In the context of our exercise, using the wave number form of angular frequency helps simplify the math when dealing with waves at atomic scales. By shifting our focus from wavelength to angular frequency or wave number, we have a more straightforward approach in proving the equivalence of sine functions with differing wavelengths but identical spatial values at the points in question. The angular frequency is an indispensable tool when analyzing waves in various forms across physical systems.