Issues with seeding a pseudo-random number generator more than once?

I've seen quite a few recommendations for not seeding pseudo-random number generators more than once per execution, but never accompanied by a thorough explanation. Of course, it is easy to see why the following (C/C++) example is not a good idea:

int get_rand() {
  srand(time(NULL));
  return rand();
}

since calling get_rand several times per second produces repeated results.

But wouldn't the following example still be an acceptable solution?

MyRand.h

#ifndef MY_RAND_H
#define MY_RAND_H

class MyRand
{
  public:
    MyRand();
    int get_rand() const;
  private:
    static unsigned int seed_base;
};

#endif

MyRand.cpp

#include <ctime>
#include <cstdlib>
#include "MyRand.h"

unsigned int MyRand::seed_base = static_cast<unsigned int>(time(NULL));

MyRand::MyRand()
{
  srand(seed_base++);
}

int MyRand::get_rand() const
{
  return rand();
}

main.cpp

#include <iostream>
#include "MyRand.h"

int main(int argc, char *argv[]) 
{
  for (int i = 0; i < 100; i++) 
  {
    MyRand r;
    std::cout << r.get_rand() << " ";
  }
}

i.e. even though MyRand:s constructor is called several times in rapid succession, each call to srand has a different parameter. Obviously, this is not thread-safe, but then again neither is rand.

Solution

Each time you call a pseudo-random number generator function, the generator takes some internal state and produces a pseudo-random number and a new internal state. The algorithm for transforming the internal state is carefully chosen so the output appears random.

When you seed the random number generator, you're basically setting this internal state. If you reset the internal state to some predictable value, you'll lose the appearance of randomness.

For example, a popular, simple RNG is a linear congruential generator. Numbers are generated like this:

X[n+1] = (a X[n] + c) mod m

In this case, X[n+1] is both the result and the new internal state. If you seed the generator every time as you suggest above, you'll get a sequence that looks like this:

{(ab + c) mod m, (a(b+1) + c) mod m, (a(b+2) + c) mod m, ...}

where b is your seed_base. This doesn't look random at all.