The speed of a sound wave depends on its wavelength `lamda`, and frequency v. Find an expression for the spped of sound.

Answer

To find the expression for the speed of sound in terms of its wavelength (λ) and frequency (ν), we can follow these steps:

### Step 1: Understand the relationship
The speed of sound (V) is related to its wavelength (λ) and frequency (ν). We can express this relationship as:
\[ V = k \cdot \lambda^a \cdot \nu^b \]
where \( k \) is a constant, and \( a \) and \( b \) are the powers to which wavelength and frequency are raised, respectively.

### Step 2: Write down the dimensions
Next, we need to consider the dimensions of each quantity involved:
- The speed of sound (V) has dimensions of length per time:
\[ [V] = L^1 T^{-1} \]
- The wavelength (λ) has dimensions of length:
\[ [\lambda] = L^1 \]
- The frequency (ν) is the reciprocal of time, so its dimensions are:
\[ [\nu] = T^{-1} \]

### Step 3: Substitute dimensions into the equation
Now we substitute the dimensions into our equation:
\[ [V] = [k] \cdot [\lambda]^a \cdot [\nu]^b \]
This translates to:
\[ L^1 T^{-1} = [k] \cdot (L^1)^a \cdot (T^{-1})^b \]
This simplifies to:
\[ L^1 T^{-1} = [k] \cdot L^a \cdot T^{-b} \]

### Step 4: Equate dimensions
To find the values of \( a \) and \( b \), we equate the dimensions on both sides:
1. For the length (L):
\[ 1 = a \]
2. For the time (T):
\[ -1 = -b \]
This simplifies to:
\[ b = 1 \]

### Step 5: Substitute values of a and b
Now that we have found \( a = 1 \) and \( b = 1 \), we can substitute these values back into our original equation:
\[ V = k \cdot \lambda^1 \cdot \nu^1 \]
This gives us:
\[ V = k \cdot \lambda \cdot \nu \]

### Step 6: Identify the constant k
In the context of sound waves, the constant \( k \) is typically taken to be 1 when we are discussing the basic relationship between speed, wavelength, and frequency. Thus, we can express the speed of sound as:
\[ V = \lambda \cdot \nu \]

### Final Expression
The final expression for the speed of sound in terms of its wavelength and frequency is:
\[ V = \lambda \cdot \nu \]

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