Calculating the observed wavelength for water waves from a moving boat

Boat is moving with 10 km/h, waves have frequency 2 Hz and wavelength 1 m.

GOOGLE GEMINI:

In physics, when an

observer (the boat) moves through a medium toward or away from a stationary source, the physical wavelength in the water does not change, but the observed (apparent) wavelength changes based on the relative speed between the boat and the wave crests. 

Calculation for 2026 

1. Calculate the wave speed (

$v_w$

):
The speed of the waves in the water is:

$v_w=f\times \lambda =2\text{Hz}\times 1\text{m}=2\text{m/s}$

2. Convert the boat's speed (

$v_b$

):
A speed of 10 km/h is approximately 2.78 m/s (

$10,000/3,600$

). 

3. Determine the observed wavelength (

${\lambda }_{obs}$

):
The observed wavelength is the distance between crests as measured by the boat. This depends on whether you are moving into the waves (increasing frequency) or with the waves (decreasing frequency). 

• Scenario A: Moving into the waves (head-on)
  The boat encounters wave crests faster. The observed frequency (
  

  $f_{obs}$
  ) is:
  
  

  $f_{obs}=f\left(\frac{v_w+v_b}{v_w}\right)=2\left(\frac{2+2.78}{2}\right)=4.78\text{Hz}$
  
  The observed wavelength (
  

  ${\lambda }_{obs}$
  ) is the distance the boat perceives between crests:
  
  

  ${\lambda }_{obs}=\frac{v_w}{\text{apparent frequency shift factor}}=\frac{v_w\cdot \lambda }{v_w+v_b}=\frac{2\cdot 1}{2+2.78}\approx 0.418\text{m}$
• Scenario B: Moving with the waves (from behind)
  If the boat is moving at 2.78 m/s, it is actually moving faster than the 2 m/s waves. It would "overtake" the crests from behind.
  
  

  ${\lambda }_{obs}=\left|\frac{v_w\cdot \lambda }{v_w-v_b}\right|=\left|\frac{2\cdot 1}{2-2.78}\right|\approx 2.56\text{m}$
   

Summary for 2026 

While the physical wavelength in the water remains 1 meter, the observed wavelength from the boat is: 

• ~0.42 m if moving directly into the waves.
• ~2.56 m if moving in the same direction as the waves (overtaking them). 

If the boat moves at

10 km/h into waves with a frequency of 3 Hz and a wavelength of 2 m, the observed frequency will be approximately 4.39 Hz with an observed wavelength of 1.37 m (if moving toward the waves), or an observed frequency of 1.61 Hz with an observed wavelength of 3.72 m (if moving with the waves). 

Step 1: Calculate Wave Speed (

$v_w$

) and Boat Speed (

$v_b$

) 

First, determine the propagation speed of the waves in the water and convert the boat's speed into meters per second:

$v_w=f\times \lambda =3\text{Hz}\times 2\text{m}=6\text{m/s}$

$v_b=\frac{10\text{km/h}}{3.6}\approx 2.78\text{m/s}$

---

Step 2: Calculate Observed Frequency (

$f^{\prime }$

) 

The frequency changes because the boat's motion relative to the waves affects how many crests it hits per second. We use the Doppler formula for a moving observer:

$f^{\prime }=f\left(\frac{v_w\pm v_b}{v_w}\right)$

. 

• Moving Toward Waves:
  

  $f_{toward}^{\prime }=3\left(\frac{6+2.78}{6}\right)\approx 4.39\text{Hz}$
• Moving With Waves:
  

  $f_{with}^{\prime }=3\left(\frac{6-2.78}{6}\right)\approx 1.61\text{Hz}$
   

---

Step 3: Calculate Observed Wavelength (

${\lambda }^{\prime }$

) 

The observed (apparent) wavelength is the effective distance between crests as perceived by the boat, calculated by

${\lambda }^{\prime }=\frac{v_w}{f^{\prime }}$

: 

• Moving Toward Waves:
  

  ${\lambda }_{toward}^{\prime }=\frac{6}{4.39}\approx 1.37\text{m}$
• Moving With Waves:
  

  ${\lambda }_{with}^{\prime }=\frac{6}{1.61}\approx 3.72\text{m}$
   

---

Answer: 

If moving toward the waves, the observed frequency is 4.39 Hz and the observed wavelength is 1.37 m. If moving with the waves, the observed frequency is 1.61 Hz and the observed wavelength is 3.72 m

Yes, for a boat moving relative to waves in a medium like water, an observer on the boat will measure the

apparent (observed) wavelength differently than its physical wavelength in the water. 

Using your data (frequency 3 Hz, wavelength 2 m, boat speed 10 km/h

$\approx$

2.78 m/s): 

• Moving toward the wave: The boat encounters wave crests more quickly, resulting in a higher observed frequency and a shorter observed wavelength of approximately 1.37 meters.
• Moving with the wave: The boat travels in the same direction as the wave, meaning it takes longer to encounter successive crests, resulting in a lower observed frequency and a longer observed wavelength of approximately 3.72 meters. 

Summary of Observed Measurements (2026) 

Direction	Observed Frequency ( $f^{\prime }$ )	Observed Wavelength ( ${\lambda }^{\prime }$ )
Toward waves	~4.39 Hz	~1.37 m
With waves	~1.61 Hz	~3.72 m

How it's calculated: 

1. Wave Speed (
   

   $v_w$
   ):
   

   $3\text{Hz}\times 2\text{m}=6\text{m/s}$
   .
2. Relative Velocity (
   

   $v_{rel}$
   ): The speed at which the boat and waves pass each other (
   

   $v_w+v_b$
   toward;
   

   $v_w-v_b$
   with).
3. Observed Wavelength (
   

   ${\lambda }_{obs}$
   ): Calculated as
   

   $\frac{v_w}{f^{\prime }}$
   , representing the distance between crests as they pass the boat's reference frame.